338 



REPORT — 1869. 



yfhevG f =fj + h ^J -I, f=cj— Its/ — I, 



Y=(M+N/)(P + Q/) 

 to tlie sum of two of the form 



dji ^ dij 



J(i+ny)VY' Jc 



method i 



where n' and n" are real. The method is foimded on substituting 



.-?/(! + ^/) ^^ r dp 



VY Jl + e/^' 



and then finding two suitable values for ^. The section to which I refer 

 is almost detached from the rest of the memoir, and may be read without 

 diificulty. There is another paper by Pdchelot on this subject in the Sith 

 volume of Crelle's Journal. 



Section 4. — Of late the theory of continued fractions has received a very 

 wide extension. A very beautiful example of this will be found in the 56tli 

 volume of Crelle's Journal, in which Laplace's coefficients are applied to the 

 expansion of functions in continued fractions. It is not, therefore, surprising 

 that the exi^ansion of elliptic functions in this manner should have been the 

 subject of several successful eflTorts. In the 39th volume of CreUe's Journal 

 there is a paper on this subject by Professor Malmsten. The known equa- 

 tions 



dk dk 



(1) 



easily lead to the differential equation 



(y'-2k"i/ + k'yUc + H-"di/=0 (2) 



E7.; 

 where 2/=|^- 



If we put 



f l+'^l i w, ^ ^ l — (c + 2mx)- 

 l-=^mx+-^^^ y=i(l-c-2»uv) + 4«7 " • • • (3) 



this last equation may be written 



(m'x"- + mcx+^{r-l))('l^^ + iA + mu(2mx + c) + lm-=0 . . (4) 

 But from (3), 



E'A- 



Fk 



m 



Therefore by the second of equations (1), 



2Jc']c'^-=. 





or 



u = 



m 

 2h 



711 



dF' 



dk ^dJr 

 "Y dv 



¥ 



(IF dF 

 dl- Jlf^ 

 F F ' 



* I bare been tho fuller here because several steps are oiiiittcd by Malmsten. 



