6 REPORT 1860. 



which are of the form (IV.) discovered by Mr. Cayley (Phil. Mag. vol. viii. 1859, 

 p. 34), who there first enumerated the forms of groups of eight. 

 Two such groups can be completed with unity, and any one of the 



„ n8 =7.6.5.3.2 



substitutions of the form &. 



It is easy to form groups of Mr. Cayley's form (II.) ; e. g., 



12345678 

 34127856 

 23416785 

 41238567 

 56781234 

 78563412 

 67852341 

 85674123 



which is-one of the grouped groups whose general theory I have handled in a memoir 

 which will shortly see the light. 



On a new Proof of Pascal's Theorem. By the Rev. T. Rennison, M.A. 



On Systems of Indeterminate Linear Equations. 

 By H. J. Stephen Smith, M.A., Fellow of Balliol College, Oxford. 

 The object of this communication was to point out the connexion which exists 

 between particular solutions of indeterminate linear equations, and their most gene- 

 ral solution. The principle upon which this connexion depends may be explained in 

 a very particular case. Let the sytem of indeterminate equations reduce itself to the 

 single equation 



Ax+By + Cz=0, (1) 



in which we may suppose A, B, C to have no common divisor ; let also a, b, c and 

 a', b', c' be two different solutions of that equation in integral numbers ; then, if the 

 three numbers 



bc' — b'c, ca' — ac', ab' — a'b (2) 



admit of no common divisor, the complete solution of the indeterminate equation is 

 contained in the formulae 



x = at+a'u, 1 



y=bt+b'u,\ (3) 



Z=ct + c'u, ) 



in which t and u are absolutely indeterminate integral numbers ; but if the condi- 

 tion (2) be not satisfied, the formula? (3) will not represent all, but only some of the 

 solutions of the equation (1). If, therefore, by any method, as for example that of 

 Euler, we have arrived at formula of the type of the formulas (3), which demonstra- 

 bly contain the complete solution of the indeterminate equation, we may be certain 

 that the three numbers analogous to the numbers (2) admit of no common divisor. 

 Thus, by applying Euler's method of solution, which is explained in most books of 

 algebra, to the indeterminate equation Ax+By-±-Cz = 0, we obtain the solution of a 

 celebrated problem, first considered by Gauss in the 'Disquisitiones Arithmetical,' 

 of which the following is the enunciation. 



" Given 3 numbers A, B, C, to find six others, 



a, b, c, 

 such that a', b', c', 



A = bc'-b'c, B=ca'-ac', C=ab' -a'b." 

 Other methods more symmetrical, and perhaps not more tedious than that of Euler, 

 were also suggested in this paper for the treatment of indeterminate equations, and 

 for the resolution of an important class of arithmetical problems which depend on 

 those equations in the manner just explained. 



