By Laenas Gifford Weld

The purpose of the writer of the current paintings has been to boost the idea of Determinants within the easiest attainable demeanour. nice care has been taken to introduce the topic in this kind of means that any reader having an acquaintance with the rules of uncomplicated Algebra can intelligently stick with this improvement from the start. The final chapters has to be passed over by way of the coed who's no longer conversant in the Calculus, and an analogous is to be stated in connection with a few few of the previous articles ; yet in no case will the continuity of the path be tormented by such omissions.

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**Sample text**

We shall suppose therefore, that X = 0 identically, and that in the expressions of the functions X , there are no terms, independent of the quantities x,, of order lower than m 1. Thus, we can assume that g = - 1, for we can achieve this by replacing x and y by x[(-"'"~-')]-' and y[(-g)'""-')]-', respectively. Under these conditions, let us assume that + + + + + Y = -x" + g1xm+1 + ... + yY1 + Y , , where Yl does not depend on the quantities x, and vanishes for x = y = O . = x, = 0. Suppose now, that y = c q(x, c) is the general integral of the equation dy/dx = Y,.

Thus, we can assume that g = - 1, for we can achieve this by replacing x and y by x[(-"'"~-')]-' and y[(-g)'""-')]-', respectively. Under these conditions, let us assume that + + + + + Y = -x" + g1xm+1 + ... + yY1 + Y , , where Yl does not depend on the quantities x, and vanishes for x = y = O . = x, = 0. Suppose now, that y = c q(x, c) is the general integral of the equation dy/dx = Y,. Here c is an arbitrary constant and ~ ( x c) , + Investigation of One of t h e Singular Cases of Theory of Stability of Motion 37 a holomorphic function of x and c, vanishing for x = 0.

1 -- @p + @ p r+ @ y r 2 + ... , R, = RjO) where R(",OF) are functions of 3, independent of r, we deduce easily from the expressions in the right-hand sides of Eqs. (27) that all Ry)for which s < q - 1 are of the form S(3)P(3), P(3) denoting a rational entire function of C(3). )for which s < q - 1 are of the form P(3). Also the V(q+'-3)forwhich I < q 2 are entire functions of the quantities di), with coefficients of the form P(3). Thus, taking into account that S(3) = - dC(3)/d9, we must , 3 ) ,...