TRANSACTIONS OF SECTION A. 451 



where Sj = ■v/(Zi' — b^) ', 



and since if 



.r, = + 6 vo -j — 



1 1 — c« It 



•'1 ^ 1 — CM V 



then u + v + iv-=0; 



and when .I'l = y^ then Sj is finite, 



yj = Sj „ i'j is finite, 



Si=-i"i . » yi is finite; 

 we see that Euler's integral may be written in the forms 



Cri+y, + =0(?A-=i)^=(Yi-ZO* 

 (.t'l + 2/i + Sj) (zj - .r^)^ = (Zi - Xj)-, 

 or adding and reducing 



Zx.y^z, = 36» + YjZi + ZjXj + XiYj. 

 This equation (since Xj Yj are rational in x and y) is, when cubed, the complete 

 integral of the difi'erential equation 



X-irf.f + Y-% = 0. 

 It will be found to be 



27 XYZ = {SDxyz + G{yz + zx + xy) + BGr + y + s) + 3A} ^ 

 The developed equation is of the form 



a + hy + g2/2 + py3 

 + X {h + by + f y'~ + qy^} 

 + .T2{g + fy'^ + cy- + ry^} 

 + .r' {p + qy^ + ry^ + dy'' } = 0. 



The coefficients, of which ten are different, are therefore derivable from a matrix 

 of the form of the determmant which expresses the discriminant of the general 

 quadric surface 



It = a.i- + by- + cs'^ + 2fys + 2gs,r + 2h.ry + 2p.r + 2qy + 2rs + d = 0. 

 (Compare Cayley, ch. xiv. p. 340, ' Elliptic Functions.') 



The result shows that the complete integral of the differential equation is an 

 equation «t = 0, where it is a symmetric cubi-cubic function of (.f, y) : that is, a sym- 

 metric function, cubic in regard to each variable separately. 



5. On the Establishment of the Elementary Principles of Quaternions on 

 an Analytical Basis.^ By Gustate Plarr, D.Sc. 



The writer is of opinion, that for the purpose of initiating the student into the 

 knowledge of the principles of quaternions, it woidd be more convenient (to many 

 minds at least) to have these principles established by the analytical method. 

 This method presents the advantage of founding the analytical properties of vectors 

 and quaternions on a clear basis, which is no other than definition, and assumption 

 by definition, and of establishing the geometrical properties on these definitions 

 by way of interpretation and deduction. This is also the method applied to the 

 treatment of the more important applications of quaternions — we mean to say, the 

 method which consists in analytical deduction followed by geometrical interpreta- 

 tion of the results. 



The working principle by which the geometrical properties may be deduced 

 will be the following: Assuming that the product of two vectors is to be effected 



' A Paper by the same author, and bearing the above title, has been communi- 

 cated by Professor Tait to the Royal Society of Edinburgh, and inserted in vol, 

 xxvii., part II. (pp. 175 to 202), of the Tranmctions of that Society. 



G Q 2 



