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LXXXIX. A Kinematic Method of finding the Hoots which 

 are less than Unity of a Rational Integral Equation of any 

 Degree : followed by a more general Method of finding the 

 Roots of any Value. By Thomas H. Blakesley*. 



I. A kinematic method of solving a rational integral equation 

 of any degree for roots less than unity. 



THE method depends mainly upon two facts, one of a 

 mathematical character, the other kinematical. 



The mathematical proposition is that every integral power 

 of cos 0, as cos" 0, may be expressed in a finite series of terms 

 involving one only of the quantities cos?i#, cosn~'Z0, 

 cos n — 4:0, &c, the last term involving cos if n is odd, and 

 not involving # ? or absolute, if n is even. 



The kinematical fact is that with an appropriate system of 

 links it is possible to make a series of points move so that 

 the polygon formed by joining them will always remain 

 equiangular, though tlie angle between its consecutive sides 

 will vary, the sides themselves being at the same time main- 

 tained in any given ratio among themselves, though not to a 

 fixed unitf. 



Suppose a given equation to be 



0=A + B# + Gc 2 -r- + N.e", 



and substitute cos for x, then 



0=A + Bcos6>-fCcos 2 6'+ + Ncos"fl. 



If now for every power of cos 6 is substituted its value in 

 cosines of multiples of there will result an equation of the 

 form 



= R + Ri cos # + R 2 cos 2(9 + + R n cosrc0, 



in which the coefficients R . . . R„ are readily obtained linear 

 functions of the coefficients AB ... X, of the original equation. 

 Now suppose an equiangular polygon formed (fig. 1), with 

 the sides proportional to R ...R, n taken in order and laid 

 end to end. Upon each side erect an isosceles triangle with 

 sides proportional to the chord which forms its base, and in 

 every case extend the sides beyond the base until the nearer 

 sides of the next-but-one triangles intersect, finishing the 

 first and last cases in an obvious way. 



* Communicated by the Author. 



t For an introduction to the ideas involved in this sort of motion, 

 vide a paper on " Logarithmic Lazytongs and Lattice-works/' in the 



Philosophical Magazine of September 1907. 



