( 187 ) 



IX. — Some Identities connected with Alternants, and with Elliptic Functions. 



By Thomas Muir, LL.D. 



(Read December 3, 1900.) 



(1) Cayley in his paper* entitled "Note sur l'addition des fonctions elliptiques" 

 obtains among other similar things an expression for 



in terms of 

 where 



S(u + v -f- w . . . ) 

 Su, Sv, Sw, . . . 



Su = Jk • sinam —pr 



The form of the expression is the quotient of two determinants, and as the expression 

 becomes useless for such cases as u = v, u = w, . . . on account of the simultaneous 

 vanishing of numerator and denominator, he is led to seek a means of throwing out the 

 common evanescent factors. In doing so there is brought to light the need for the 

 existence of certain identities, viz., 



in connection with the numerator and denominator for the case of three arcs the 

 respective identities 



1 



a A 



1 



b B 



1 



c C 



1 



a aA 



1 



b 6B 



1 



c cC 



(B + C)(C + A)(A + B) = 



(B + C)(C + A)(A + B) = 



1 



a 



A 2 



1 



b 



B 2 



1 



c 



C 2 



1 



a 



aA 2 



1 



b 



6B 2 



1 



c 



cC 2 



(A 2 +B 2 +C 2 + BC + CA + AB) - 



(A 2 + B 2 + C 2 +BC+CA+AB) - 



1 a A 4 



1 b B* 



1 c C 4 



1 a aA i 



1 b 5B 4 



1 c cC 4 



in connection with the numerator for the case of four arcs 



1 



a a 2 aA 



1 



b ¥ bB 



1 



c c 2 cC 



1 



d d* dD 



(A + B)(A + C)(A + D)(B + C)(B + D)(C + D) 



1 



a 



a 2 



aA 2 



M - 



1 



b 



W 



Z>B 2 





1 



c 



c 2 



cC 2 





1 



d 



d 2 



dB 2 





1 



a a 2 



aA 4 



N + 



1 



b b 2 



Z>B 4 





1 



c <? 



cC 4 





1 



d d 2 



dD 4 





a a 2 



aA 6 



b b 2 



&B 6 



G C 2 



cC 6 



d d 2 



dD 6 



p, 



* Crelle's Journ., xli. pp. 57-65 ; or Collected Math. Papers, i. pp. 540-549. 

 VOL. XL. PART I. (NO. 9). 



2 E 



