284 6. B. AIRY, ESQ. ON A PROOF OF THE THEOREM 



Substituting these new quantities in the equation, it becomes 



where P = af cos {nd + a) + 6r"~' cos |(n - l) + j3| + + m cos fx, 



Q = ar' sin {nd + a) + ir""' sin {(« - 1) + /3} + + m sin ju. 



And the problem of discovering a root of the equation is reduced to this, To find values for 

 r and 6 which shall make P and Q simultaneously = 0. 



2. The theorem of proving the existence of a root of the equation will therefore be 

 reduced to this : It is to be shewn that, upon trying all values of r between and positive 

 infinity, and all values of Q between and Ztt, there will certainly be at least one value of r 

 and one value of Q which, used in combination, will make both P and Q = 0. It will be 

 found unnecessary to make trial of negative values of r, because (the equation being of an 

 even order) they will give the same results in the first terms of P and Q as the positive 

 values, and because, in the demonstration, their effects on the other terms will be unimportant. 

 Moreover, in every term of P and Q, the effect of a change of sign of r may also be produced 

 by retaining the sign of r and altering the value of by ir; a change which will merely 

 produce results contained among those now to be mentioned. 



3. It will be convenient to confine our attention, in the first instance, to values of 9 

 included between those which make nQ + a = 0, nQ + a = Stt. But the same form of demon- 

 stration which applies to these will also apply to values 



between those which make nO + a = Stt, nd ■¥ a = ^ir ', 

 between those which make nd + a = 47r, n0 + a = 67r ; 

 and so on, ending with SwTr, after which the values of recur. 



After the value mr (w being necessarily even, and nir therefore necessarily occurring as a 

 limit of values of nO + a), the values of 9 recur increased by the quantity ir. The substi- 

 tution of a certain value p for r with these values of 9 amounts to exactly the same as the 

 substitution of — p for r with the former values of 9; and thus the absence of all necessity for 

 trial of negative values of r, to which allusion has been made above, is confirmed. 



4. If we construct two curves, whose common abscissa is 9, and whose ordinates are the 

 corresponding values of P and Q produced by substituting in their expressions the same 

 value of r ; and if we vary the value of r ; then, upon increasing indefinitely the value of r, 

 we shall increase indefinitely, and to an unmanageable magnitude, the ordinates representing 

 P and Q. But, as all that we want in the subsequent demonstration depends upon the 

 proportion of P and Q, we can adopt a device, similar to that introduced so successfully by 

 Newton in the 1st Section of the Principia, but in the opposite sense ; we can suppose every 

 one of the ordinates for a given large value of r diminished in the same proportion, till the 



