4 io SCIENCE PROGRESS 



proof of the propositions contained in the two previous sub- 

 sections ; but it has the merits of being brief and of dealing 

 with both propositions together. 



If possible, let k be a number such that <f>k = k. Then 

 obviously </>'& = k, <f> 3 k = k, . . . and (f>"k = k. 



Next, find the successive tangentials of <ji v k. By the 

 ordinary rule 



dfpk = Wfi'- 1 . </>>' ! ~ ! . <t>'<f> 71 - 3 . ..<t>'4> • <f>']k 

 = (4>'k)\ 



To form the second tangential take logarithms of both sides 

 of the first of these equations, and differentiate. Hence 



^ t 4> n k = (<i>'k) u - l .(f>' , k.{i +cf>'k + (c/>'&) 2 + {<t>'ky . . .}. 



The subsequent tangentials must therefore all contain the 

 factor ((f>'k) n multiplied by groups of geometric progressions 

 in <f>'k affected by <\>"k, $"'k, . . . and so on. Now if 4>'k is 

 numerically less than unity, (</>'&)" becomes infinitely small 

 when n is made infinitely large. Hence as §'k, (f>"k, §'"k . . . 

 are supposed finite, and the geometric progressions referred to 

 tend to finite limits when <f>'k is numerically less than unity, 

 it follows that, under these circumstances, (<f>"Yk, (<f> n )"k, ($ n )'"k 

 ... all vanish. Thus by Taylor's Theorem 



tf>*(k + h) = k ; 



at least if h is not too large. This is the equation of a straight 

 line parallel to the axis of x and passing through the point 

 (k, k), h being a length along x. 



Now let the continuous curve / have a succession of real 

 roots Xi, X-2, X z . . . Then the curve /' must be alternately 

 positive and negative at these roots. Let it be negative at 

 X lf X Zf X 5 . . . ; and let <£ = o +/. Then at all the points 

 X 1 ,X 2 ,X 3 . ..,<f>X = X; thatis,</> w X = X; that is, [o +f]'X=X. 

 But, as just seen, we have also [o + f] n (X -\- h) = X if n is very 



large and if-r-[o + f]X is numerically less than unity; that is, 



if [i + f'~\X is less than i and greater than — i ; that is, if f'X 

 is negative and greater than — 2. Putting x = X -f h, we 

 therefore have [0 +/]°°#o =X, if f'X is negative and greater 

 than — 2. 



But this condition can hold only for the roots X X) X z , X 5 . . ., 



