4860 



Integration by Definite Integrals 



4860 



. y = 0, and other equations. Bras- 



sinne, E. Toul. Mm. Ac. 2 (1846) 326-; 

 3 (1847) 204- ; 1 (1863) 483-. 



z"- 1 K + b n x) y (n] + xn ~ 2 K-i + V-i*) 2/ ( "-" + - 



+ (a 1 + b 1 x)y' + b y = 0. 

 Malmsten, C. J. Crelle J. 39 (1850) 99-. 



y" x + y-* + Ax m y = Q. Malmsten, C. J. Crelle 



J. 39 (1850) 108-. 

 f IT (a r + b r x p ) x( m - r > y( m ~ r 1 = S c r x^. Most, E. 



T= Z. Mth. Ps. 15 (1870) 427-7 

 X y" + ky'-y = Q. Le Paige, C. M. M. H. H. 

 Brux. Ac. Bll. 41 (1876) 1011-. 



Cockle, (Sir) Jas. QJ. Mth. 17 (1881) 

 293-. 



x y4 + j- + y ( x + -A) = 0, from theory of re- 

 flexion of sound. SJtarpe, H. J. Mess. Mth. 

 10 (1881) 174-. 

 d^y dy . ., _ d cos (x + 1) , 



theory of sound. Sharpe, H. J. [1881] Mess. 

 Mth. 11 (1882) 41-. 



^!| _ (ae* + 2) ^ + y = 0. Florov, P. S. (xn) 



Kharkov Mth. S. Com. (1883) 127- ; Fschr. 

 Mth. (1884) 288. 



Heymann, W. Z. Mth. Ps. 28 (1883) 

 241 -. 



*^ =:r m tt . Fiorov, P. S, Kharkov Mth. S. 

 dx n 



Com. (1885) 131- ; Fschr. Mth. (1886) 302. 



= *. Florov, P. S. Kharkov Mth. S. 

 df l 

 Com. (1887) 81- ; Fschr. Mth. (1887) 337-. 



4860 Integration of ordinary linear 

 equations by definite integrals. 

 (See also 4430.) 



Mainardi, G. A. Sc. Lomb. Yen. 7 (1837) 

 185-. 



Wantzel, L. C. E. 17 (1843) 1191-. 



Spitzer, S. Schlomilch Z. 3 (1858) 224-; 

 4 (1859) 37-. 



Stem, A. [1868-80] Sk. Nf . F. 10 (1869) 101- ; 

 Kjob. Ov. (1880) 237-. 



Pincherle, S. Bologna Ac. Sc. Mm. 2 (1891) 

 523-. 



Schlesinger, L. C. E. 120 (1895) 1396-. 



Bilinear relations between hypergeometric in- 

 tegrals. Hirsch, A. Mth. A. 52 (1899) 

 130-. 



Cases. Lobatto, E. Crelle J. 17 (1837) 



Cases. Hansted, B. (xn) Ts. Mth. 4 (1880) 



121-. 

 Differential and difference equations with 1 or 



more independent variables. Brajtzew, J. E. 



Fschr. Mth. (1900) 356. 

 Equation, 2nd order. Hoppe, E. Grunert 



Arch. 27 (1856) 55-. 

 , , with linear coefficients. Thomae, J. 



Z. Mth. Ps. 14 (1869) 349-. 

 with rational coefficients, relations due to 



definite integrals. Mellin, H. Helsingf. 



Acta 21 (1896) No. 6, 57 pp. 

 Hypergeometric equation. LStnikov, A. V. 



Eec. Mth. (Moscou) 11 (*1883-84) 327-; 



Fschr. Mth. (1884) 273-. 

 Integrable equations. Nekrasov, P. A. Rec. 



Mth. (Moscou) 15 (1891) 277-, 608- ; Mth. 



A. 38 (1891) 509-. 

 Integral, general, of 



+ A n y=f(x), 



formula for, by multiple definite integral. 



Aoust, . C. R. 96 (1883) 775-. 

 Integration by definite integrals of product of 



2 arbitrary functions, cases. Boussinesq, J. 



C. E. 94 (1882) 33-. 

 , method. Petzval, J. Haidinger Ab. 1 



(1847) 177-. 

 Eeduction of integrals of equations, 1st order, 



in mixed differences, to simple definite in- 

 tegrals. Grosso, (abbate) E. del. [1852] (xn) 



Nap. Ac. Pont. At. 6 (1854) 323-. 

 Eiccati's equation. Kummer, E. E. Crelle J. 



12 (1834) 144-. 

 Simultaneous equations. Mellin, H. Acta Mth. 



22 (1899) 41-. 

 Solutions of systems of equations by means 



of multiple integrals, Jacobi's formulae. 



Vasillev, A. V. Kazan S. Nt. (Ps.-Mth.) P. 



4 (1886) 39-; Fschr. Mth. (1886) 264-. 

 Spitzer's priority claims. Petzval, J. [1857] 



Wien SB. 28 (1858) 253-. 

 Tissot's equation. Pochhammer, L. Mth. A. 



37 (1890) 512-. 

 Total differential equations. Du Bois-Eey- 



mond, P. [1868] Crelle J. 70 (1869) 



299-. 



^- = x m y. Kummer, E. E. Crelle J. 19 

 dx n 



(1839) 286-. 

 sy" + (r + q.r) y'+ (p + nx + wix 3 ) y = 0. Spitzer, 



S. Grunert Arch. 23 (1854) 121-. 



x mgj =2/> Spitzer, S. C. E. 48 (1859) 



746- ; Crelle J. 57 (1860) 82-. 

 x m^. = ay , Spitzer, S. C. E. 48 (1859) 



995-. 

 y'"=xy'-ny. Spitzer, S. Schlomilch Z. 7 



(1862) 113-. 

 x*y'"=y. Spitzer, S. Schlomilch Z. 8 (1863) 



292-. 

 yW = Ax*y"+Bxy'+ Cy. Spitzer, S. [1870] 



Mth. A. 3 (1871) 453-. 



388 



