STEWART'S THEOREMS. 119 



and therefore to 



Now A" 1 = 2a m sin mn$ + S6 m cos mnfa (m integral) . 

 Hence /(<) + . . . +/( < + - - 2?r J = S m sin rarc< + &c. 



Let the index of the highest power of sin <j> or cos <j> in f<j> 

 be p ; then it is easily seen that when /</> is developed, as it may 

 always be, in a series of sines and cosines of the multiple arcs, 

 p$ will be the largest arc that can enter into the development. 

 But if n is greater than p, mn<j) will be greater than p<f>, except 

 when m is zero. Hence the development 



2a m sin mn<j> + &c. 



cannot coincide with that obtained by summing the separate 

 developments of /<, f( <j> + ) , &c., unless a m and b m are = 



\ 71 / 



in every case, except when m = 0. Hence as sin mn<f) = when 

 m = 0, the expression will be reduced to 5 , and we shall have, 

 when n>p, 



+ .../(< + ^ - 2?r) = b Q ... 



a constant. 



Q. E. D. 



The constant 1> will of course be the sum of the constant 

 parts of the developments of /<, &c. ; and as these are all equal 

 and are n in number, it will be n times the constant term in f<f>. 



1 f 2ir 

 Now by Fourier's theorem this is equal to I f<j> . d(j), as in- 



27T J Q 



deed is obvious. Hence 



which is our fundamental formula. 



The first of Stewart's propositions is the following : 



From any point in the circumference of a circle draw per- 

 pendiculars p, p v &c. to the sides of a regular w-sided polygon 

 circumscribed about it ; then, if r is the radius, 



3 = 5wr 8 ...... (1) ...... w>3. 



