342 REPORT — 1869. 



\ dr dr J 



then V, considered as a function of v and X, satisfies the partial differential 

 equation 



d\ dv^ ^ ^ ^ ^ ^ ^ dv'' 



Let y=l, a=ft=^, and the differential equation becomes 



dY _d.\' _f?'V_n 

 d\ du dv' 



and T becomes 



Jo ^'-U-Oa-'-O ^J o"^^l-'-siii>' 



where <=sin"^. 



From this Jacobi shows that 



^"•V? 



Ae 



/ Eurf«-i-— - 

 *' K 



satisfies the partial differential equation, originally seen to be tme from the 

 nature of the series 



^ dq ~ d.v^ ■ 



Section 4. — Jacobi's third paper is a very remarkable one, and there is a 

 French translation in Liouville's Journal. It is entitled " IJeber die Differ- 

 entialgleichung welcher die Ileihen l±2q + 2q'^± . . . 2' ^/q + 2W f~+ . . . 

 Genuge leisten." 



The maiu object of the paper is to show that 



-\/?' -^/¥' "-VW- 



each satisfy the following differential equation. 



By differentiating the equation 



p^ dd 



K= I ' Vl-Psin-'tf 

 Jo 



with respect to P it is proved without much difficulty that if y=A^—, 



d.PJc^'Ll. „ 



dh' - 4 • ^■^'' 



