1 74 ^- FUJISAWA. 



on the opening page of the present paper. I was not aware of the 

 existence of Jacobi's formulae when I first found the formula for 

 snmt which was, in fact, then obtained in a sUghtly different form 

 as furnished by (17.), and which is perhaps in some respects preferable 

 to the one here given. I have, however, given it its present shape, 

 in order that, for the particular values of //, it may exactly coincide 

 with the formula} given by the illustrious mathematician whose 

 memory is sacred to every student of the theory of elliptic functions. 



§. 9. 



Before proceeding further with the reduction of the multiplica- 

 tion-formulas, it is necessary to give a few formula? relating to the 

 differentiation of composite functions, to which frequent reference will 

 subsequently be made. 



Let u be a function of y and y a function of x. It is required to 

 find the n"' differential coefficient of a with respect to x. By actual 

 differentiation, we find 



du _ du dt/ 

 dx dy dx ' 



dPu 

 dx 



I _ dit d-y d-n. / di/ \- 

 '' ~ ~dy W' "^ 111/' \dx) ' 



(Pa _ dt(_ d^ d-u dy d-y d^it / dy V 



dx' ~ dy Ix'^'dj/'l^'dx''^ ~dJf \dx) ' 



dx' dy dx' "^ dy' \ ^ dx dx' "^ ^ V dx') S "^ *' dy' \W) W 



.dhi/dyV 

 ■^ d,/\dxj' 



