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Celebratio Mathematica

Gilbert Ames Bliss

Calculus of variations  ·  U Chicago

Gilbert Ames Bliss: 1876–1951

by L. M. Graves

The con­tri­bu­tions made by Gil­bert Ames Bliss to math­em­at­ics and to edu­ca­tion­al and sci­entif­ic activ­it­ies in the United States were many and var­ied. The fol­low­ing is a quite in­ad­equate sum­mary of his life and work, with spe­cial ref­er­ence to his math­em­at­ic­al activ­it­ies. Gil­bert Bliss was born in Chica­go on May 9, 1876, and died in In­galls Me­mori­al Hos­pit­al in Har­vey, Illinois, on May 8, 1951. In 1893, one year after the Uni­versity of Chica­go first opened its doors to stu­dents, he en­rolled there as a stu­dent, and was awar­ded the de­grees of B.S. in 1897, M.S. in 1898, and Ph.D. in 1900. He stud­ied at the Uni­versity of Göttin­gen dur­ing the year 1902–1903. His first teach­ing was done as a sub­sti­tute for a mem­ber of the staff of Kala­ma­zoo Col­lege for sev­er­al weeks dur­ing the year 1898–1899. He was an in­struct­or in math­em­at­ics at the Uni­versity of Min­nesota 1900–1902, an as­so­ci­ate at Chica­go 1903–1904, as­sist­ant pro­fess­or at the Uni­versity of Mis­souri 1904–1905, pre­cept­or (\( = \) as­sist­ant pro­fess­or) at Prin­ceton 1905–1908, and at Chica­go was as­so­ci­ate pro­fess­or 1908–1913, pro­fess­or 1913–1933, Mar­tin A. Ry­er­son Dis­tin­guished Ser­vice Pro­fess­or 1933–1941, and chair­man of the de­part­ment 1927–1941. He re­tired from act­ive ser­vice in 1941. In vari­ous sum­mer or au­tumn terms from 1906 to 1911 he gave courses at Wis­con­sin, Chica­go, Prin­ceton, and Har­vard.

Dur­ing his stu­dent days Bliss first fell un­der the in­flu­ence of F. R. Moulton, then a young as­sist­ant who was teach­ing at Chica­go, and his first pub­lished pa­per was en­titled The mo­tion of a heav­enly body in a res­ist­ing me­di­um. The fel­low­ship in as­tro­nomy for which he ap­plied was not gran­ted, and he even­tu­ally de­cided to de­vote him­self to pure math­em­at­ics. His thes­is for the M.S. was en­titled The geodes­ic lines on the an­chor ring, and his thes­is for the Ph.D. bore the same title. The first de­veloped ex­pli­cit for­mu­las for the geodes­ies in terms of el­lipt­ic in­teg­rals and dis­cussed some of their prop­er­ties. In the Ph.D. dis­ser­ta­tion [1] it was shown that the points on the in­ner equat­or of the an­chor ring are of the first kind (in the clas­si­fic­a­tion due to Man­goldt), i.e., each geodes­ic passing through such a point con­tains no con­jug­ate point, while all oth­er points on the an­chor ring are of the second kind, i.e., not of the first kind. Pre­vi­ous in­vest­ig­a­tions by Jac­obi and by Man­goldt had shown that on a sur­face of neg­at­ive curvature all points are of the first kind, while on a closed sur­face of pos­it­ive curvature all points are of the second kind. Dur­ing his time as a gradu­ate stu­dent Bliss made a copy of Bolza’s re­cord of Wei­er­strass’ 1879 course on the cal­cu­lus of vari­ations. This re­cord to­geth­er with Bolza’s in­flu­ence un­doubtedly helped fix in Bliss’ mind the in­terest in the sub­ject which dom­in­ated his re­search. Dur­ing his stay at Min­nesota he stud­ied Kneser’s book on the sub­ject, which was the first prin­ted ex­pos­i­tion of Wei­er­strass’ ideas, and pub­lished a pa­per [2] on the second vari­ation and suf­fi­cient con­di­tions for a min­im­um when one end point is vari­able. A second pa­per [4], tak­ing up the case when both end points are vari­able, was writ­ten dur­ing his stay in Göttin­gen. In both these pa­pers, as in his thes­is, geo­met­ric con­sid­er­a­tions play a prom­in­ent role. There are fre­quent evid­ences of his con­tinu­ing in­terest in geo­metry. In his pa­pers [7], [8], [17], he dis­cussed “Finsler” geo­metry of two di­men­sions, where the arc length is giv­en by an in­teg­ral of the form \[ \int f(x,y,\tau)(x^{\prime\mkern1mu 2} + y^{\prime\mkern1mu 2})^{1/2}dt,\\ \quad \tan \tau = \frac{y^{\prime}}{x^{\prime}}. \] Finsler’s thes­is treat­ing the \( n \)-di­men­sion­al case did not ap­pear un­til 1918. The tensor ana­lys­is of such spaces was de­veloped still later by Élie Cartan. Bliss kept in touch with these ideas, and de­vel­op­ments of them ap­peared in the dis­ser­ta­tions of Taylor [e6], John­son [e14], House­hold­er [e21], and Stokes [e25], which he sug­ges­ted and su­per­vised. Dur­ing the years 1908–1910, im­me­di­ately fol­low­ing the death of Masch­ke, he lec­tured on geo­metry at Chica­go, but when Bolza re­turned to Ger­many in 1910, Bliss seemed glad to make ana­lys­is again the fo­cus of his teach­ing.

A num­ber of pa­pers ap­ply­ing the meth­ods of Wei­er­strass to a vari­ety of prob­lems in cal­cu­lus of vari­ations ap­peared from Bliss’ pen dur­ing the years be­fore the first world war. Some of these were writ­ten in col­lab­or­a­tion with Max Ma­son [6], [9], [11] and one was a joint pa­per with A. L. Un­der­hill [16]. An idea which has be­come ba­sic for much sub­sequent work in the cal­cu­lus of vari­ations ap­peared in the pa­per [18]. This is the con­sid­er­a­tion of the min­im­um prop­er­ties of the second vari­ation. It makes pos­sible simple proofs of the ne­ces­sity of the Jac­obi con­di­tion, and of the ana­log­ous con­di­tion for the more gen­er­al prob­lem of Bolza, and is a guid­ing prin­ciple in the con­struc­tion of suf­fi­ciency proofs. It led to the pa­per [30] on the trans­form­a­tion of Cleb­sch which ex­presses the second vari­ation in the form \[ I_2(\eta) = \int^{x_2}_{x_1} F y^{\prime}_{i}y^{\prime}_{k} (\eta^{\prime}_{i} - \pi_i)(\eta^{\prime}_{k} - \pi_k)\, dx. \] This for­mula is val­id when there ex­ists a con­jug­ate sys­tem of ac­cess­ory ex­tremals whose de­term­in­ant does not van­ish on the in­ter­val \( [x_1, x_2] \) and the vec­tor func­tion \( \pi \) is suit­ably defined in terms of the ad­miss­ible vari­ation \( \eta \) (van­ish­ing at \( x_1 \) and \( x_2 \)) and of the giv­en con­jug­ate sys­tem. It is de­rived very simply with the help of the the­ory of fields of ex­tremals and the for­mula of Wei­er­strass. A simple dir­ect proof is also giv­en. The meth­ods of this pa­per are ba­sic for the vari­ous suf­fi­ciency proofs sub­sequently giv­en for the prob­lems of Lag­range, May­er, and Bolza.

Dur­ing the years 1916–1946 Bliss de­voted much time to im­prov­ing and ex­tend­ing the the­or­ies of the prob­lems of Lag­range, May­er, and Bolza. In this he had the co­oper­a­tion of sev­er­al of his stu­dents, not­ably Hestenes. The res­ults ap­peared in defin­it­ive form in 1946 in the second part of his “Lec­tures” [38] where the prob­lem of Bolza is treated in de­tail. If one com­pares this with earli­er ex­pos­i­tions of these gen­er­al prob­lems — for ex­ample, of the prob­lem of Lag­range in Bolza’s Vor­le­sun­gen, or in Bliss’ pa­per [31] — one is im­me­di­ately im­pressed with the great­er scope of the the­ory, due to weak­en­ing of hy­po­theses both for ne­ces­sary con­di­tions and for suf­fi­cient con­di­tions, and with the sim­pli­fic­a­tions ob­tained in the proofs. The first part of the “Lec­tures” con­tains an un­usu­ally clear present­a­tion of the the­ory of the cal­cu­lus of vari­ations for cases when there are no side con­di­tions.

Al­though Bliss pub­lished only one pa­per on mul­tiple in­teg­rals in the cal­cu­lus of vari­ations [37] he began to dis­cuss the sub­ject in courses and sem­inars in the 1920’s, and con­tin­ued at fre­quent in­ter­vals up to the sum­mer of 1942. Cor­al, Cour­ant, Mc­Shane, Radó, and Smi­ley were among those from out­side the Uni­versity of Chica­go who par­ti­cip­ated in some of these sem­inars. Bliss ex­pounded the sub­ject of mul­tiple in­teg­rals in mi­meo­graphed lec­ture notes [33], [35], where vari­ous im­prove­ments in the the­ory may be found. Some of his ideas were de­veloped in the doc­tor­al dis­ser­ta­tions of Sim­mons [e7], Cor­al [e16], Raab [e18], Cosby [e19], Nord­haus [e23] and Landers [e24]. One of the out­stand­ing prob­lems in this do­main is to find con­di­tions en­sur­ing the ex­ist­ence of a field suit­able for use in a suf­fi­ciency proof.

Dur­ing the 1920’s Bliss also lec­tured on bound­ary value prob­lems as­so­ci­ated with the cal­cu­lus of vari­ations, and on ap­plic­a­tions to quantum mech­an­ics and re­lativ­ity. Some of his work on bound­ary value prob­lems ap­peared in the pa­pers [28], [29], [32], [36]. For the sake of the greatest sym­metry and gen­er­al­ity it is con­veni­ent to study dif­fer­en­tial sys­tems of the form \begin{align} \label{one} & \frac{dy}{dt}=[A(x)+ \lambda B(x)]y,\\ & My(a) + Ny(b)=0. \notag \end{align} (Here cap­it­al let­ters are used to de­note square matrices of or­der \( n, y \) and \( z \) de­note matrices of one column, and the trans­pose of \( A \), for ex­ample, is de­noted by \( A^{\prime} \).) The ad­joint sys­tem to \eqref{one} is defined to be \begin{align*} & \frac{dz}{dt} = -z^{\prime} [A+\lambda B],\\ & z^{\prime}(a)P+z^{\prime}(b)Q=0, \end{align*} where \( MP - NQ = 0 \). The sys­tem \eqref{one} is said to be self-ad­joint if it is equi­val­ent to its ad­joint un­der a trans­form­a­tion \( z = T(x)y \) where the mat­rix \( T \) is nonsin­gu­lar on the in­ter­val \( [a, b] \). In the pa­per [36], a self-ad­joint sys­tem \eqref{one} was defined to be def­in­ite in case: (a) the mat­rix \( T^{\prime}B \) is sym­met­ric and pos­it­ive semi­def­in­ite, and (b) the sys­tem \eqref{one} has no non­trivi­al solu­tions for which \( B_y = 0 \). The bound­ary value prob­lems arising from prob­lems of Bolza sat­is­fy­ing cer­tain mild re­stric­tions are of this type. The defin­i­tion of “def­in­itely self-ad­joint” giv­en in the earli­er pa­per [29] ex­cluded many such prob­lems. However, in both pa­pers prop­er­ties of the sys­tem are de­rived which are like those which hold for the Fred­holm equa­tion with real sym­met­ric ker­nel. Stu­dent theses re­lated to the sub­ject of bound­ary value prob­lems in­clude those of Miss Stark [e8], Bam­forth [e10], Cope [e11], Hick­son [e15], Miss Jack­son [e12], Hu [e17], and Miss Wig­gin [e20]. The in­verse prob­lem of the cal­cu­lus of vari­ations drew some at­ten­tion from Bliss, al­though it was not in the main line of his in­terest. (See the pa­per [10]. Among his stu­dents Dav­is [e9], La Paz [e13], and Mo­scov­itch [e22] con­trib­uted to the dis­cus­sion of the prob­lem.

Cal­cu­lus of vari­ations the­ory re­quires the use of ex­ist­ence the­or­ems for im­pli­cit func­tions and dif­fer­en­tial equa­tions which yield more in­form­a­tion than those com­monly giv­en in the text­books and treat­ises on ana­lys­is. Hence Bliss was led to write a num­ber of pa­pers on the sub­ject [4], [5], [13], [19] and his Col­loqui­um lec­tures de­livered be­fore the Amer­ic­an Math­em­at­ic­al So­ci­ety in 1909 [15] were de­voted to these and re­lated top­ics. The ana­lys­is of sin­gu­lar points for trans­form­a­tions of the plane was one of the top­ics treated in the Col­loqui­um. This work grew out of spe­cial cases arising in the cal­cu­lus of vari­ations. The dis­ser­ta­tion of Lovitt [e2] treated some cases for trans­form­a­tions of three-space. The trans­form­a­tions of the plane con­sidered in the Col­loqui­um are real ana­lyt­ic ones, and real branches of plane ana­lyt­ic curves play a prom­in­ent role. This led to the two pa­pers on the factor­ing of power series [12], [14], and later to the study of al­geb­ra­ic curves [25], [26], [34]. At the date of these stud­ies the geo­met­ric proofs for the the­or­ems on the re­duc­tion of the sin­gu­lar­it­ies of al­geb­ra­ic curves were quite un­sat­is­fact­ory. In re­cent years mod­ern al­gebra has provided the means for a very ab­stract and gen­er­al treat­ment of the the­ory of al­geb­ra­ic curves. Bliss gave a clear treat­ment from the view­point of ana­lys­is of the case when the base field is the com­plex num­ber field. This phase of his work in­dic­ates that he was not al­ways in­ter­ested in the max­im­um ab­stract­ness, al­though he al­ways sought for sim­pli­city, clar­ity, and com­pre­hens­ive­ness in his math­em­at­ic­al ex­pos­i­tion.

Upon the ur­ging of Veblen in 1918 Bliss went to Ab­er­deen as a sci­entif­ic ex­pert in the Range Fir­ing Sec­tion. His work on the dif­fer­en­tial cor­rec­tions of tra­ject­or­ies [20], [21], [22] made pos­sible enorm­ous sav­ings in the time re­quired to com­pute the range cor­rec­tions which are ne­ces­sary in al­low­ing for the ro­ta­tion of the earth, ef­fects of wind, and vari­ations in the dens­ity of the air, powder charge, etc. His meth­ods con­tin­ued to be used dur­ing World War II. While the ad­vent of high speed com­put­ing ma­chines has led to some changes in meth­ods, the ba­sic the­or­et­ic­al ideas re­main the same. These ba­sic ideas were ex­pounded in two pa­pers in Trans. Amer. Math. Soc. [24], [23], and later in a book [3]. Bliss’ in­terest in func­tion­al ana­lys­is was not con­fined to this peri­od. In earli­er years the dis­ser­ta­tions of Fisc­her [e1], Lam­son [e3], Le Stour­geon, [e4] and Barnett [e5] were re­lated to this field.

Bliss was known the world around as one of the lead­ing au­thor­it­ies on the cal­cu­lus of vari­ations, al­though he did not con­trib­ute to the new dir­ec­tions of study opened up by Ton­elli and by Morse. He was primar­ily in­ter­ested in math­em­at­ic­al re­search, but he heeded also the call of oth­er du­ties. His broad in­terest in math­em­at­ics was evid­enced, for ex­ample, by his reg­u­lar at­tend­ance at the meet­ings of the Math­em­at­ic­al Club of the Uni­versity of Chica­go, where he con­trib­uted com­ments and ques­tions on a wide vari­ety of top­ics. He felt the duty of ex­pos­i­tion, and wrote the first of the series of Carus Mono­graphs, for which he also served as a mem­ber of the ed­it­or­i­al com­mit­tee. In math­em­at­ic­al pub­lic­a­tion he took a middle ground between those who pour forth un­di­ges­ted ideas and those who in­sist on ex­cess­ive pol­ish­ing. In 1925 he was the re­cip­i­ent of the first award of the Chauven­et Prize by the Math­em­at­ic­al As­so­ci­ation of Amer­ica for his pa­per Al­geb­ra­ic func­tions and their di­visors [27]. Bliss was an as­so­ci­ate ed­it­or of the An­nals of Math­em­at­ics from 1906 to 1908, and of the Trans­ac­tions of the Amer­ic­an Math­em­at­ic­al So­ci­ety from 1909 to 1916. He was chair­man of the Ed­it­or­i­al Com­mit­tee of the Uni­versity of Chica­go Sci­ence Series from 1929 un­til his re­tire­ment. From 1924 to 1936 he served on the Fel­low­ship Board of the Na­tion­al Re­search Coun­cil. He was a trust­ee of the Teach­ers In­sur­ance and An­nu­ity As­so­ci­ation for sev­er­al years. A wide vari­ety of or­gan­iz­a­tions called on him to make in­form­al ad­dresses, and as a res­ult he gave quite a num­ber of such talks on math­em­at­ics and re­lated top­ics.

He was elec­ted to the Na­tion­al Academy of Sci­ences in 1916, to the Amer­ic­an Philo­soph­ic­al So­ci­ety in 1926, and was made a Fel­low of the Amer­ic­an Academy of Arts and Sci­ences in 1935. Dur­ing his term (1921 and 1922) as pres­id­ent of the Amer­ic­an Math­em­at­ic­al So­ci­ety, he con­duc­ted a cam­paign for new mem­bers, in co­oper­a­tion with E. R. Hedrick, Chair­man of the Mem­ber­ship Com­mit­tee. As a res­ult the mem­ber­ship of the So­ci­ety in­creased from 770 to 1127 dur­ing this bi­en­ni­um. In 1930 he was Vice Pres­id­ent and Chair­man of Sec­tion A of the Amer­ic­an As­so­ci­ation for the Ad­vance­ment of Sci­ence. He was also a mem­ber of the Math­em­at­ic­al As­so­ci­ation of Amer­ica, the Illinois Academy of Sci­ence, the Lon­don Math­em­at­ic­al So­ci­ety, the Deutsche Math­em­at­ische Ver­ein, and the Cir­colo Matem­atico di Palermo. In 1935 the Uni­versity of Wis­con­sin awar­ded him the hon­or­ary de­gree of Doc­tor of Sci­ence.

Bliss played a cent­ral role in the plan­ning of Eck­hart Hall, which houses the De­part­ment of Math­em­at­ics at Chica­go, and he closely su­per­vised its erec­tion. It is very largely due to his care and in­sight that this build­ing fills so well the needs of the De­part­ment, and it will stand as a beau­ti­ful and en­dur­ing me­mori­al to his ef­forts. He ac­cu­mu­lated a rather sub­stan­tial per­son­al lib­rary of math­em­at­ic­al books and peri­od­ic­als, which he presen­ted to the De­part­ment of Math­em­at­ics in 1950 for the use of the staff. This “Gil­bert Ames Bliss Lib­rary” is now housed in the Fac­ulty Con­fer­ence Room in Eck­hart Hall. It in­cludes an un­usu­ally com­plete col­lec­tion of works on the cal­cu­lus of vari­ations.

Bliss be­lieved whole­heartedly in the im­port­ance of teach­ing as an ac­com­pani­ment of math­em­at­ic­al re­search. All those who ex­per­i­enced the stim­u­la­tion of his courses re­tain a high ad­mir­a­tion for him as very nearly the ideal sci­entif­ic teach­er. His judg­ment was highly re­garded, and his coun­sel was sought from many quar­ters, by oth­er uni­versit­ies as well as his own, and by oth­er or­gan­iz­a­tions be­sides the math­em­at­ic­al ones. He was con­ser­vat­ive in tend­ency, and was not one to rush in­to new do­mains of math­em­at­ics or in­to new pro­jects in war time. In his view fun­da­ment­al math­em­at­ic­al re­search and teach­ing were activ­it­ies too im­port­ant to be in­ter­rup­ted ex­cept for very strong reas­ons. At­tempts to re­strict stu­dent en­roll­ments by high ad­mis­sion re­quire­ments seemed to him un­wise, since some poorly trained stu­dents de­vel­op real in­tel­lec­tu­al power, and some bril­liant ones gradu­ally fade. With re­spect to the value of the Ph.D. train­ing, he stated:

The real pur­pose of gradu­ate work in math­em­at­ics, or in any oth­er sub­ject, is to train the stu­dent to re­cog­nize what men call the truth, and to give him what is usu­ally his first ex­per­i­ence in search­ing out the truth in some spe­cial field and re­cord­ing his im­pres­sions. Such a train­ing is in­valu­able for teach­ing, or busi­ness, or whatever activ­ity may claim the stu­dent’s fu­ture in­terest.

Gil­bert Bliss was mar­ried to Helen Hurd in 1912. Their chil­dren are Eliza­beth (Mrs. Rus­sell Wiles), born in 1914, and Ames, born in 1918. His wife was stricken and died in the in­flu­enza epi­dem­ic of Decem­ber, 1918. In 1920 he mar­ried Olive Hunter, who sur­vives him. The Blisses had a sum­mer home in Flossmoor, Illinois, for many years, and be­gin­ning in 1931 they made their year round res­id­ence in Flossmoor, where Gil­bert was at one time a mem­ber of the Vil­lage Board of Trust­ees and head of the Po­lice Com­mis­sion. They were ex­cep­tion­ally friendly in man­ner and in spir­it, with a high sense of hu­mor, and en­joyed en­ter­tain­ing stu­dents, fac­ulty col­leagues, and friends in their home. Golf at one of the nearby coun­try clubs was a fa­vor­ite re­cre­ation un­til de­clin­ing health forced its dis­con­tinu­ance. Bliss was in­ter­ested in com­pet­it­ive sports throughout his life, and in his earli­er years par­ti­cip­ated act­ively in bi­cycle ra­cing, ten­nis, and rac­quets.

His in­flu­ence on math­em­at­ics and math­em­aticians was wide­spread and deep, and his con­tri­bu­tions will be long re­membered.

[Ed­it­or’s note: This art­icle was ori­gin­ally pub­lished with a full bib­li­o­graphy of Bliss.]

Works

[1]G. A. Bliss: The geodes­ic lines on the an­chor ring. Ph.D. thesis, Uni­versity of Chica­go, 1900. Ad­vised by O. Bolza. MR 2936782

[2]G. A. Bliss: “The second vari­ation of a def­in­ite in­teg­ral when one end-point is vari­able,” Trans. Amer. Math. Soc. 3 : 1 (1902), pp. 132–​141. MR 1500591 JFM 33.​0385.​01

[3]G. A. Bliss: “Jac­obi’s cri­terion when both end-points are vari­able,” Math. Ann. 58 : 1–​2 (1903), pp. 70–​80. MR 1511229 JFM 34.​0402.​01

[4]G. A. Bliss: “An ex­ist­ence the­or­em for a dif­fer­en­tial equa­tion of the second or­der, with an ap­plic­a­tion to the Cal­cu­lus of Vari­ations,” Trans. Amer. Math. Soc. 5 : 2 (1904), pp. 113–​125. MR 1500665 JFM 35.​0340.​01

[5]G. A. Bliss: “The solu­tions of dif­fer­en­tial equa­tions of the first or­der as func­tions of their ini­tial val­ues,” Ann. of Math. (2) 6 : 2 (January 1905), pp. 49–​68. MR 1503546 JFM 36.​0389.​01

[6]G. A. Bliss and M. Ma­son: “A prob­lem of the Cal­cu­lus of Vari­ations in which the in­teg­rand is dis­con­tinu­ous,” Trans. Amer. Math. Soc. 7 : 2 (1906), pp. 325–​336. MR 1500752 JFM 37.​0402.​01

[7]G. A. Bliss: “A gen­er­al­iz­a­tion of the no­tion of angle,” Trans. Amer. Math. Soc. 7 : 2 (1906), pp. 184–​196. MR 1500741 JFM 37.​0490.​01

[8]G. A. Bliss: “A new form of the simplest prob­lem of the Cal­cu­lus of Vari­ations,” Trans. Amer. Math. Soc. 8 : 3 (1907), pp. 405–​414. MR 1500795 JFM 38.​0408.​01

[9]M. Ma­son and G. A. Bliss: “The prop­er­ties of curves in space which min­im­ize a def­in­ite in­teg­ral,” Trans. Amer. Math. Soc. 9 : 4 (1908), pp. 440–​466. MR 1500821 JFM 39.​0441.​01

[10]G. A. Bliss: “On the in­verse prob­lem of the Cal­cu­lus of Vari­ations,” Ann. of Math. (2) 9 : 3 (April 1908), pp. 127–​140. MR 1502361 JFM 39.​0445.​01

[11]G. A. Bliss and M. Ma­son: “Fields of ex­tremals in space,” Trans. Amer. Math. Soc. 11 : 3 (1910), pp. 325–​340. MR 1500866 JFM 41.​0437.​01

[12]G. A. Bliss: “A new proof of Wei­er­strass’s the­or­em con­cern­ing the fac­tor­iz­a­tion of a power series,” Bull. Amer. Math. Soc. 16 : 7 (1910), pp. 356–​359. MR 1558920 JFM 41.​0286.​03

[13]G. A. Bliss: “A new proof of the ex­ist­ence the­or­em for im­pli­cit func­tions,” Bull. Amer. Math. Soc. 18 : 4 (1912), pp. 175–​179. MR 1559180 JFM 43.​0483.​04

[14]G. A. Bliss: “A gen­er­al­iz­a­tion of Wei­er­strass’ pre­par­a­tion the­or­em for a power series in sev­er­al vari­ables,” Trans. Amer. Math. Soc. 13 : 2 (April 1912), pp. 133–​145. MR 1500910 JFM 43.​0504.​02

[15]G. A. Bliss: “Fun­da­ment­al ex­ist­ence the­or­ems,” pp. i–​ii,1–​107 in The Prin­ceton Col­loqui­um (Prin­ceton Uni­versity, 15–17 Septem­ber 1909), part 1. Amer­ic­an Math­em­at­ic­al So­ci­ety (New York), 1913. JFM 44.​0365.​01

[16]G. A. Bliss and A. L. Un­der­hill: “The min­im­um of a def­in­ite in­teg­ral for uni­lat­er­al vari­ations in space,” Trans. Amer. Math. Soc. 15 : 3 (1914), pp. 291–​310. MR 1500981 JFM 45.​0605.​03

[17]G. A. Bliss: “Gen­er­al­iz­a­tions of geodes­ic curvature and a the­or­em of Gauss con­cern­ing geodes­ic tri­angles,” Amer. J. Math. 37 : 1 (January 1915), pp. 1–​18. MR 1507894 JFM 45.​0859.​02

[18]G. A. Bliss: “Jac­obi’s con­di­tion for prob­lems of the Cal­cu­lus of Vari­ations in para­met­ric form,” Trans. Amer. Math. Soc. 17 : 2 (April 1916), pp. 195–​206. MR 1501037 JFM 46.​0758.​03

[19]G. A. Bliss: “Solu­tions of dif­fer­en­tial equa­tions as func­tions of the con­stants of in­teg­ra­tion,” Bull. Amer. Math. Soc. 25 : 1 (1918), pp. 15–​26. MR 1560139 JFM 47.​0940.​01

[20]G. A. Bliss: “A meth­od of com­put­ing dif­fer­en­tial cor­rec­tions for a tra­ject­ory,” J. U.S. Ar­til­lery 51 (1919), pp. 445–​449. Cor­rec­ted copy of earli­er print­ing in vol. 50 (pp. 455–460).

[21]G. A. Bliss: “The use of ad­joint sys­tems in the prob­lem of dif­fer­en­tial cor­rec­tions for a tra­ject­ory,” J. U.S. Ar­til­lery 51 (1919), pp. 296–​311.

[22]G. A. Bliss: Dif­fer­en­tial cor­rec­tions for anti-air­craft guns. Technical report, Ab­er­deen Prov­ing Ground, 1919.

[23]G. A. Bliss: “Func­tions of lines in bal­list­ics,” Trans. Amer. Math. Soc. 21 : 2 (1920), pp. 93–​106. MR 1501138 JFM 47.​0382.​02

[24]G. A. Bliss: “Dif­fer­en­tial equa­tions con­tain­ing ar­bit­rary func­tions,” Trans. Amer. Math. Soc. 21 : 2 (1920), pp. 79–​92. MR 1501137

[25]G. A. Bliss: “Bira­tion­al trans­form­a­tions sim­pli­fy­ing sin­gu­lar­it­ies of al­geb­ra­ic curves,” Trans. Amer. Math. Soc. 24 : 4 (1922), pp. 274–​285. MR 1501226 JFM 50.​0266.​02

[26]G. A. Bliss: “The re­duc­tion of sin­gu­lar­it­ies of plane curves by bira­tion­al trans­form­a­tion,” Bull. Amer. Math. Soc. 29 : 4 (1923), pp. 161–​183. Pres­id­en­tial ad­dress de­livered be­fore the Amer­ic­an Math­em­at­ic­al So­ci­ety, Decem­ber 28, 1922. MR 1560693 JFM 49.​0264.​01

[27]G. A. Bliss: “Al­geb­ra­ic func­tions and their di­visors,” Ann. of Math. (2) 26 : 1–​2 (September–October 1924), pp. 95–​124. MR 1502680 JFM 50.​0699.​03

[28]G. A. Bliss: “A bound­ary value prob­lem in the Cal­cu­lus of Vari­ations,” Bull. Amer. Math. Soc. 32 : 4 (1926), pp. 317–​331. MR 1561219 JFM 52.​0509.​02

[29]G. A. Bliss: “A bound­ary value prob­lem for a sys­tem of or­din­ary lin­ear dif­fer­en­tial equa­tions of the first or­der,” Trans. Amer. Math. Soc. 28 : 4 (1926), pp. 561–​584. MR 1501366 JFM 52.​0453.​13

[30]G. Bliss: “The trans­form­a­tion of Cleb­sch in the Cal­cu­lus of Vari­ations,” pp. 589–​603 in Pro­ceed­ings of the In­ter­na­tion­al Math­em­at­ic­al Con­gress (Toronto, 11–16 Au­gust 1924), vol. 1. Edi­ted by J. C. Fields. Uni­versity of Toronto Press, 1928. JFM 54.​0532.​03

[31]G. A. Bliss: “The prob­lem of Lag­range in the Cal­cu­lus of Vari­ations,” Amer. J. Math. 52 : 4 (October 1930), pp. 673–​744. MR 1506783 JFM 56.​0435.​01

[32]G. A. Bliss and I. J. Schoen­berg: “On sep­ar­a­tion, com­par­is­on and os­cil­la­tion the­or­ems for self-ad­joint sys­tems of lin­ear second or­der dif­fer­en­tial equa­tions,” Amer. J. Math. 53 : 4 (October 1931), pp. 781–​800. MR 1506854 JFM 57.​0528.​01 Zbl 0003.​25702

[33]Top­ics of the Cal­cu­lus of Vari­ations. Edi­ted by G. Bliss. Uni­versity of Chica­go, 1932. Prin­cip­ally a re­port on lec­tures by E. J. Mc­Shane and Richard Cour­ant.

[34]G. A. Bliss: Al­geb­ra­ic func­tions. AMS Col­loqui­um Pub­lic­a­tions 16. Amer­ic­an Math­em­at­ic­al So­ci­ety (New York), 1933. JFM 59.​0384.​03 Zbl 0008.​21004

[35]G. A. Bliss: The Cal­cu­lus of Vari­ations, mul­tiple in­teg­rals. Uni­versity of Chica­go, 1933. MR 2936825

[36]G. A. Bliss: “Def­in­itely self-ad­joint bound­ary value prob­lems,” Trans. Amer. Math. Soc. 44 : 3 (1938), pp. 413–​428. MR 1501974 JFM 64.​0438.​02 Zbl 0020.​03204

[37]G. A. Bliss: “The Cal­cu­lus of Vari­ations for mul­tiple in­teg­rals,” Amer. Math. Monthly 49 : 2 (February 1942), pp. 77–​89. MR 0006022 Zbl 0063.​00455

[38]G. A. Bliss: Lec­tures on the cal­cu­lus of vari­ations. Uni­versity of Chica­go Press (Chica­go, IL), 1946. MR 0017881 Zbl 0063.​00459