381 



In the foregoing operations, we have analyzed the interval [0, 1], 

 within which the four roots are all comprised, by commencing with the 

 roots of the quadratic X % = 0, seeing that these roots are real, and their 

 values so easily determined. There is, of course, no necessity to reach 

 this quadratic through the descending steps at p. 368 ; we may ascend 

 to it by means of the three leading coefficients 4 - 10 + 9 ; thus the 

 simple equation is 4 x 4x - 10 = 0, that is, \6x - 10 = 0; and the 



quadratic, 3 * 16 ff 2 + 3x1 Ox + 9 = 0, that is, 24a 2 - SOx + 9 = 0. But 

 if, without regarding this quadratic, we had commenced with the lead- 

 ing figure of the root of the simple equation, namely, ~ ~ = *6 . . . , 



a proceeding which would have been in strict accordance with the ge- 

 neral directions at p. 370, the corresponding transformed equation would 

 have been 



4# 4 - -4x* - -36# 2 + -454;r + -0484 * 0 ; 



so that the original interval would then have been subdivided into the 

 two partial intervals [0, -6] and [*6, 1], each comprising two roots; 

 and the character of each pair would become known by contracting each 

 of these intervals as above. 



Note 3. 



It is here proposed to prove that whenever the condition of imagi- 

 nary roots holds or fails for any triad of the functions X p , X^, X^ 2 , 

 &c, as deduced from the primitive X 0 , for an assigned value of x, it 

 will in like manner hold or fail, for the same value of x, for the cor- 

 responding triad (the first, second, third, &c.) when X p is taken for the 

 primitive function. 



If we take X x for the new primitive, the series of expressions fur- 

 nished by X x and its derivees will be the original series X t , X 2 , X z , &c, 

 multiplied respectively by 1, 2, 3, 4, 5, &c, taken in order. 



If X 2 be taken for the new primitive, X 2 and its derivees will be the 

 original X 2 X 3 , Xt, &c, multiplied, in order, by 1, 3, 6, 10, 15, &c. 



And if X 3 be taken for the new primitive, X 3 and its derivees will 

 be obtained by multiplying the original X 3 , X 4 , X 5 , &c, by 1, 4, 10, 

 20, 35, &c, taken in order, and so on. 



This will be readily seen to be the case from a mere inspection of 

 the several developments. 



Now the general expression for the m th term in any one of these 

 series of figurate numbers is 



m(m + 1) (m + 2) (m + S) . . . (m+p-l) 



1. 2. 3. 4...p ° ' 1 J ' 



