112 REPORT— 1870. 



From these expressions it is easUy seen that the ordinary formulae rela- 

 tive to the transformation of elliptic functions may be deduced. This has 

 been done by Betti (A. D. M. 4. 26 & 57). Many of the results are, of 

 course, as must be the case in a systematic treatise, among those exceedingly 

 well known. 



Section 11. — "Wo have already given in section 11, Part II. of the present 



report, the expressions deduced by Meyer for -j- and -j-, also in section 8 



of our present division, y-. To these may be added the following (A. D. M. 

 4. p. 64), in which Y„ is what tj becomes when 7c is transformed into Ag.: — 



rnog.VV,/ 



A<r Yo-Afl 



dq 42^q-!r'-' 



eV K^ 



d\os,\/ X^A, (Y^+A,)A„ 



rZlog^V A, 



dq 4pqw'' ' 



(Y^+XvA^)A^ 



dq ^pqit"- 



Section 12. — In the papers contributed by Jacobi to the earlier numbers 

 of CrcUc's Journal, several propositions may be found which are not con- 

 tained in the ' Fundamenta Nova.' One of the most celebrated of those 

 has been the subject of a special memoir by Professor Cayley*. It is this. 

 If u= s/lc sin am z, and 



._ ?LrJ ?t«=-i+AiM"'-3 + Aw»^-5H (— 1) - u 



\hBinam2iz=(—l) li u. — . , 



1 + A^u + A.^u'+ ...(-])- 



-mt- 



also a=Jc+j^, then the denominator of this expression will satisfy the dif- 

 ferential equation : 



(l-mr+M0^-|-(n^-l)(aM-2«3/^ + n20i^-lKU=2«=(a=-4/^. 



A demonstration of this proposition has also been given by Betti (A. D. M. 

 4. 32), and another proposition given by Jacobi will be found at p. 13 of the 

 same volume. 



Section 13. — Since the publication of the ' Fundamenta Nova ' the third 

 elliptic integral has been discussed by Jacobi in his memoir " Sur la Eotation 

 d'un Corps," in the 39th volume of Crellc's Journal, by Betti, ' Annali di 

 Matematica,' iii. 309, and by Schellbach, ' Lehre von den Elliptischcn Inte- 

 gralen,' p. 217. 



* See also another paper by Professor Cayley, in wliich this subject is i 

 la multiplication des fonctions elliptiques," Crello, xxnx. p. 16. 



introduced, "Sur 



