218 



Mr. Graves on Cubic Equations. 



ambiguous p^ may have either of its two values, provided it 

 retain at any one time the same meaning in both places of its 

 occurrence in formula (3.). 



My resuli is presented in the following formulae, in which 

 V' and ^ denote real roots not negative : coSg"^ denotes the 

 smallest cos"^ not negative : and i denotes any term of the 

 arithmetical series — 2, — 1, 0, 1, 2, reckoned from inde- 

 finitely backward and forward. 



Let 



Qi = -^^^Vpc^+A^+x~ 



(4.) 





I 



Rl = ^J=f[ ^Qs + Q'4+ V'Q3-Q4)cOs|-L (2ix + COSo-l Q5) I 



R, = -^( J^'Q7+Q4- V'Q^Q4 ) sin 4' X(^^^+*^0«0~^QOT 



■(70 



(8.) 



5i = 2Qi^(xiOt + Av) +A(xv-An;^)^ 



52 = 2Q/(jciM, + Av)--A(jcv-X|*) [ 



53 = 2 Qi^ (jc V -X jOt) - A (x /x + A v) 

 5^ = 2 Q/ (x V — A fi) + A (x )«, + A v) 



X = V^ + a/'^ Vq 



Then 



^1 n R j_ ^2 ^Q^Rj 



^^= v1?^^''^^77. 



(9.) 

 (10.) 



(11.) 



From the preceding formulae it would be easy, did the size 

 of the page permit, to write out at full length a solution for x 

 in immediate terms of x, A, ju,, v. The relations infer se of the 

 mediate functions employed are very remarkable. It will be 

 seen that Sj, ^g, % and s^ are wanted merely as sigti-indicafors. 

 The critical cases where a sign-indicator becomes = 0, are ira- 



