ON ELLIPTIC AND HYPERELLIPTIC FUNCTIONS. 1-15 



To calculate these nine direction cosines in terms of function 6 may be 

 regarded as the leading physical object of this memoir. They are connected 

 with the angles 6, f, \p by the well-known equations : 



a =cos sin <p Bini^+ cos cos i^, 

 a' =cos sin cos »//— cos cp sin \p, 

 a"=— sin0sin0, 

 /3 = cos cos ^ sin 1^ — sin cos »//, 

 /3' = cos cos cos il^ + sin sin ^, 

 /3"=— sine cos ^, 

 ^ y =sin0sinv/', 



y =sin0cos;//, 

 y"=cos0. 



Then if A, B, C be the moments of inertia round the principal axes, wo 

 have 



Ap«+B2HCr^=7i 1 



Ay+BY+cv=^^/ ^ ^ 



dt A^^j^ + By ^ 



— =— sinflsin^, - =— sin0cos0, — = cos0 . . . . (3) 



i t t 



It follows from this that we arc able to write : 



P^-S/ JT^^^''''^^' 



P-Ch 

 A(A-C)" 



/P-Ch . ^ 

 ?= VB(B=Cr^^' 



V C(A-C) VC(A-C) 



\/(A-B)(E;t-P) 



where $ is a subsidiary angle, and ^= 



V(B-C)(A7t-Z^) 



Substituting these values of ^, 2,r in B-r^ +(A— C) rp=o, and [putting 



V(B-C)(A/i— Z^) 

 tt= nowhere n= -===: ■ , we have 



Vabc 



^ '^^ X. 7 \/(A-B)(B;t-r) 



aM= , where k=. — : so that 



Vl-Psin^^ V(B-C)(A;i-Z'-) 



1^1- -{Ml 



M = ; cos am M, 



^ VA(A-C) 



l2 



