223 
1888 - 89 .] Dr T. Muir on the Theory of Determinants. 
equations in n unknowns is taken up. The method followed is 
essentially the same as Scherk’s. 
MAINARDI (1832). 
[Trasformazioni di alcune funzioni algebraiche, e loro uso nella 
geometria e nella meccanica. Memoria di Gaspare Mainardi. 
44 pp. Pavia, 1832.] 
In his preface Mainardi explains that the algebraical functions 
referred to in the title are “ funzioni risultanti o determinanti.” 
But although he thus speaks of them as if they were known to 
mathematicians by name, and mentions the researches of Monge, 
Lagrange, Cauchy, and Binet in regard to them, he does not take 
for granted that his reader has a knowledge of any of their pro- 
perties. The one theorem on determinants, — the multiplication- 
theorem, — which forms the basis of the whole memoir, is con- 
sequently sought to be established without the use of any previously 
proved theorem. The attempt, as might be expected, is interesting. 
The first two sections (pp. 9-29) of the three into which the 
memoir is divided may be passed over without much comment. 
The first deals with the multiplication-theorem for two determinants 
of, the 2nd order, and with those applications of it to geometry 
which arise on making the elements of each determinant the 
Cartesian co-ordinates of two points in a plane. No proof is con- 
sidered necessary for this simple case, the opening paragraph of the 
memoir being ; — 
“ Rappresentate con x m x nf x aJ x b ; y m , y n , y ai y b otto 
quantita qualsivogliano, ed indicati per brevita il binomio 
X m- X a + Vm-Va COl simbolo (x m X a ) , 
il binomio 
Xn'Xb + Vn-Ub con (x n x b ) 
e simili, si provera facilmente essere 
^ ( x mVn - x n y m )(x a y b - x b y a ) 
= (Va)( X n X b) ~ {X m X b )(x n X a )P 
All the seven other paragraphs are geometrical. 
