414 Professor Hamitron on Conjugate Functions, 
number a increases constantly and continuously from 0 to a certain constant positive 
number = (7 being a certain number between 3 and 4,) the function sin a increases 
with it (constantly and continuously) from 0 to 1, but cosa decreases (constantly and 
continuously) from 1 to 0; while a continues to increase from = to m, sina decreases 
from 1 to 0, and cos a from 0 to —1; while a increases from 7 to om sin a decreases 
from 0 to —1, but cos a increases from —1 to 0; while a still increases from Sr to Qr, 
sin a increases from —t to 0, and cosa from 0 to 1, the sum of the squares of the 
cosine and sine remaining always = 1; and that then the same changes recur in the 
same order, having also occurred before for contra-positive values of a, according to 
this law of periodicity, that 
cos (a+2iz)=cosa, sin (a+2i7)=sina, (132.) 
? denoting here (as elsewhere in the present paper) any positive whole number. But 
because the proof of these well known properties may be deduced from the equations 
(132.), without any special reference to the theory of couples, it is not necessary, and 
it might not be proper, to dwell upon it here. 
It is, however, important to observe here, that by these properties we can always 
find (or conceive found) an indefinite variety of numbers a, differing from each other by 
multiples of the constant number 27, and yet each having its cosine equal to any one 
proposed number 3,, and its sine equal to any other proposed number f,, provided 
that the sum of the squares of these two proposed numbers (3,, 3,, is = 1; and reci- 
procally, that if two different numbers a both satisfy the conditions 
cosa = B,, sina = f,, (134. } 
P, and , being two given numbers, such that /3,’ + B,?=1, then the difference of these 
two numbers a is necessarily a multiple of 27. Among all these numbers a, there will 
always be one which will satisfy these other conditions 
a> —T, a; 7, (135.) 
t 
and this particular number a may be called the principal solution of the equations 
134.), because it is always nearer to 0 than any other number a which satisfies the 
same equations, except in the particular case when B,= —1, 3,=03; and because, in 
this particular case, though the two numbers 7 and —7 are equally near to 0, and 
both satisfy the equations (134.), yet still the principal solution 7, assigned by the 
conditions (135.), is simpler than the other solution —7, which is rejected by those 
last conditions. It is therefore always possible to find not only one, but infinitely 
many number-couples (m, a), differing from eaeh other by multiples of the constant 
