ON ELLIPTIC AND HYPERELLIPTIC FUNCTIONS. 337 



Multiply this by -. '^^" ' - -^fev, and siibtract from the last equation, 

 and we have by definition 



P(,.,.,.«., = (l + o')J^l,(l±l'P(0,.,<.)+lz^P(c,-,»,0} . (4) 



But (Hymers's ' Integral Calculus/ p. 284) 



l±^ n(c, n, d)- ^-^ n{c,-m, e)=- r(c, e) 



n m m 



V mn 

 From this we obtain 



.tan 



>J mn sin cos 



}• 



l±^P(e,n,0)-LjL'P(c,-m,0) = -^tan-'(VniSin0O; • (5) 

 therefore we have from (-1) and (5), remembering (2), 

 ^|(l + H)(l+0}p(c,»,0)=i^|(l + .,)(^l+^-!^|p(.„n,,0O 



-fitan-^CV nosing,); 



whence it is easy to see that if d.^ and n.^ &c. are formed from 0^ and n^ as 0j and 

 «j were formed from and n, we have 



a/ I (1 + ") (^+ ^) 1 P(oi0)=i tan-i( VjT; sin0^) 



+ 1 tan-' (Vn., sin 0,) + ^ tan-' ( V^ sin 0,) + 



This very beautiful theorem applies to the third elliptic integral with cir- 

 cular parameters ; but it is obvious that it may be easily extended to loga- 

 rithmic parameters, as has been done by Mr. Newman. In a subsequent 

 part of his paper, Mr. Newman points out the connexion between his own 

 researches and some of Jacobi's discoveries. 



Section 3. — In the 36th volume of Crclle*s Journal there is a Memoir by 



the late Professor Plana, of Turin, on the reduction of 1—7=, to elliptic 



functions, where 



T CMC ^C" "^ H + HV^ H-HV"^ 



T=G + &a + & •'- + i + (K+K'V— l).v+l-CK-KV^> 



and X=.i-'-rX.t'HA^'' + B.r+C. 



It would be impossible to give an analysis of this memoir without repro- 

 ducing it, which is the less necessary as so much has been written on this 

 subject. I shall therefore confine myself to directing the reader's attention 

 to a particular portion of it. 



At the beginning of the ninth section there is a method for reducing the 

 sums of two elliptic functions of the form 





7Y 



