150 Proceedings of the Eoyal Irish Academy. 



Startinor with the fomivila 



p\u. =pi(, -tjpii- iz.pii - £j -f J pu - £3 .pu - £1 + Jpu - ti.pu- £,;* 



0iM =Jpi-a - £1 = JFTfjTr^TWj, = JeTTToT^ : \ 



^. =J©-5-0.^ + ^: I (12) 



From the addition theorem (4), since 



^ =J¥^": «' = #'- - ^'^ « - h'' - ^-, 





or 



(in + Oi'l = - Ji- - 6 . ip - f). \ a ,j , . (-; 7i ^ 



^^ '. -^' . •' j g (1;, ^ ^,) = jjj/, + 0.^-0. 



Similarly 



rL (hi H- a.,) =JxL-B.r^ i,. I ^^iu+w^]=J^|,^e.^- ^.J 



a, (iu + b>i) = - i JiL - 6 .i> + 6 1 



4> (l^i -T- (1)3) = i . Jip - <h . <j> ^ U )■ (15) 



.t (|« - -3) = - Jj/» - ^ . 6 - (9 i 



* This fonnnla can be readily derired from Hermite's transfoimation (Greenhill, p. 130). 

 If - = - — ^ , where Oi = 4** — gtz — ffz, roots ei , e;, e3, and ^:, is the Hessian of n„ regarded 



fl; dx 



as a qaartac — ^ = — — . or, if s = pu, the roots of the qnartic in j-, ffx + ^Q^ = are 



If we solve this qnartic, which is 



I* - 4=1^-1 |jsi*-!-(2y3^-^i)x+ ^-^r<73 = 0, («) 



lb 



bv Enter's method, assuming 



X = s +\/ti + Jfe, + ^tZ ti. It. U 

 are the roots of Enler's cnbic 



<5 + 3S(^ + (3/r' - J/) « - — = 0. 



the roots of which are 



s — <5-^ — <3, z — et.z — e\, z — ei.z — ei. 

 .: the roots of (a) are 



s ± Vz — «! . i - «3 ± Vj — ea . 2 - ei, ± v'; _ <ri . J - «. 



The signs of thesqnare roots are determined by making 1/ ->- 0, thus /-iu -> ■, . pn -^ — ; 



.•. the sqoare roots are all positive for the root pl>i. 



