224 SCIENCE PROGRESS 



The quotient, therefore, agrees with the second example in 

 sub-section (3), and the given equation now becomes : 



cy l + (b — 2cp)y + (a — bp + cp % ) = o. 



The rule differs from that of algebraic division only in the 

 point that each term of the quotient operates on the whole 

 divisor instead of being multiplied into it. To find the first 

 term of the quotient, we ask what function of operating on 

 the first term of the divisor, namely 0, will convert it into the 

 first term of the dividend. The answer is evidently co J . This 

 term of the quotient now operates on the whole divisor, and, 

 subtracting the result, we have the second dividend. To 

 obtain successive terms of the quotient we proceed in exactly 

 the same way. The last term is easiest of all ; because a 

 number affected by o°, that is unity, merely reproduces itself 

 when it operates on anything. 



Again, suppose that we wish to " centre " an equation — 

 that is, to transform it so that the sum of the roots of the 

 transformed equation and therefore the coefficient of the second 

 term shall be zero. For this purpose we transfer the origin 

 by 1 /nth. of the coefficient of the second term of the original 

 equation with its sign changed. Example x s — ^x l — 2x -f 5 =0. 



— 1] o s — 30 8 — 20 + 5 [o* — 50 + 1 

 t _ 30 2 j. 3 Q _ ! 



-50+6 



-50 + 5 

 1 



Thus in this case the centred equation is y* — $y + 1 = o. 1 



As in algebraic division the successive terms of divisor and 

 dividend must both be in descending or both in ascending 

 order. In the former case we have " descending division " ; 

 in the latter case " ascending division." If the first term of 

 the divisor is simply the first power of 0, the division may be 

 called " simple." If not, the amateur will perhaps find out 

 for himself how to carry it out, even before he reads the rest 

 of this paper ! Many more examples are given in my " Verb 

 Functions," where operative division is also used for separating 

 an equation into operative factors (which can sometimes be 



1 When the divisor contains a fraction, it should be dealt with as shown in 3 (s)> 

 Example N. 



