(263) 



A SET OF LINEAR EQUATIONS CONNECTED WITH 

 HOMOFOCAL SURFACES. 



By Thomas Mum, LL.D. 



(Read February 28, 1906.) 

 1. It is usually stated that the set of equations 



1 ' 1 — + + — = 1 



&i-A b x -fi 2 b z -fi 3 b I -[3 il 



rp rp /y* rp 



6 2 -flj b 2 - )3 2 b 2 - fl 3 b 2 -fi n 



rp rp rp rp 



+ ir^7T + ir- i 7r + -v + 



b n -Pz K-p 2 b n -j3 3 b n -p n 



was first solved by Binet * in 1837 ; and it is certainly true that 

 subsequent to that date additional solutions were given by Chelini, 

 Cauchy, Hadenkamp, and Liouville. t The solution given in Tod- 

 hunter's " Theory of Equations," where a reference is misleadingly 

 made to Grunert's " Archiv," is really Liouville's. 



The object of the present note is to make known a solution which, 

 though simpler than any of those mentioned, is not brought forward 

 wholly on that account, but also in order to draw pointed attention 

 to a paper of Murphy's in which is given a general mode of dealing 

 with equations of the type above exemplified, and which bears a 

 date five years prior even to Binet's paper. J 



* Binet, J., " Observations sur des theoremes de geometrie . . ." (Journ. (de 

 Liouville) de Math., ii., pp. 248-252). 



f Chelini, D., " Formazione e dimostrazione della formula che da . . ." 

 (Giornale Arcadico lxxxv. (1840), pp. 3-12). 



Cauchy, A., " Memoire sur les fonctions alternees . . ." (Exercices d'analyse 

 et de phys. math., ii. (1841), pp. 151-159). 



Hadenkamp, " Ueber Transformation vielfacher Integrale "(Crelle's Journ., 

 xxii. (1841), pp. 184-192). 



Liouville, J., " Sur une classe d'^quations du premier degre " (Journ. (de 

 Liouville) de Math., xi. (1846), pp. 466-467). 



I Murphy, R., " On Elimination between an Indefinite Number of Unknown 

 Quantities" (Trans. Cambridge Philos. Soc, v. (1832), pp. 65-76). 



