90 



Of bis so-called universal solution I will consider only that of the 

 sixth degree. He assumes that— 



x« + mx« + nx-i + bxS -f px^ + tx + q = 



He then puts 



m2 Ao m2 



r m-'n m-* 1 Bt 



(2) P-L7B2--B3j=:S = y==+y^i-^^- 



(3) (i==yzw 



(4) oA3 — 2mnA2 + 2m"A — m-' = 



(5) tB< — mpB^ -+- in^nB — mS = 0. 

 From (4) and (5) find A and B 



Then x, y, z are found from 1, 2. 3 by means of a cubic equation. 



The author incidentally remarks that the proper combination of the 

 three values of A, and the four values of B are easily determined by 

 a little practice. The author also says that it is evident that by com- 

 paring coefficients the values of 1/a, 1/b, 1/c can be obtained. The 

 novice will find some difficulty in doing it. The real point of difficulty, 

 however, is that we have eight imknowu quantities, viz., a, b, c, x, y. z, 

 A, B, and nine equations to be satisfied, viz., five by equating coefficients, 

 and four from (1) and (2). So that the boasted solutiou is after all only 

 a solution when there is some condition placed on the roots. 



Graphic Methods in Elementary Mathematics. 

 By Robert J. Aley. 



The Automatic Temperature Regulator. 



By Chas. T. Knipp. 

 (Published in the Physical Review, VoL XII, No 1, January, 1901.) 



