102 Proceedings of the Rofjal Irish Academy. 



and two othei-s. Or 



FxP7, - VBpi, = 2^,p, 



V\fr/z — VdpCt = 24hp, 



y^fm - VSp^ = 2<i^, 

 on making obvious substitutions. From the first 



0'(VyiBi) = <^r/i, xX^yiSi) = *iSi, or S6,c-/iC, = S\y.y.c . 

 The relations simplify if 0, = 1. 



23. If aiSpipip -f a^Spipip -3- UiSpijisp = 0, and if .^a-^a.Oi = 0. we must have 

 Sp<f>,p = = Sp4y~p = Sp^hp. Hence, the values of p for which the expression 

 xipp) vanishes are the common Eolutions of these three equations. If 

 Sp(ifn - xd>.)p = touches '8p(^ - y<j»3)p = 0, two solutions are eqtiaL 



24. If there is a relation Saiasaz = the number of roots is the number 

 common to jS'/)(^, + ai^)p = and Sp{4h -^ ou'^p = for here aj = Oia, + ff»ai- 

 For a relation between the &'s we get results similar to these latter. 



25. If xCpp) ^^^ ^ reduced to the form VdipO^, the axes of 0r-&; 

 (or 6f^9i) are solutions for which x(pp) = 0. 



26. The general problem of inveimon is a difficult one. Suppose 

 <T = aiSptpip -r- a,Spa_o — a^Spdysp, we have Si = Spd>ip: Si = Sp<S>2p; SpA^ = i%, 

 where 



The complete solution depends on the solution of these equations. For the 

 particular forms ^p = p -f aSap, ©^ = p ^ /3S/3p, 4hp = p -^ y^yp> ^^ g®t 

 Si = p~ -r S^ap. Ss = p" + S'pp, Si^ p" -i- Sryp, 



and so pSa^ = V^y/lS^-p- ^ VyayS, - p- - Va^yS, - p-. 

 It p' = — X, we get here a quartic for .r. If x' is a solution 

 - x'S-a^y = { FiSYv/Si^-T' ± VyaysTT^' ± Va^y S - 

 for some combination of sign, and for the same combination 



± p'Sa(iy = V^yS^ ^ x' X Vfay/S. -h x' ± Vajiy S^ + x. 



The values of p' are, of course, equal and opposite in pairs. The inversion of 

 the general binomial form is exactly similar to the process here. 



27. If x(/>/») = Fi9,p05p = (7, we have at once p = vF&'it&Vt To get x, 

 substitute in first equation 



