LOOPS OF EPICYCLOIDAL CURVES. 123 



6. If the arms turn in opposite directions, we have to change the sign of /3 in 

 the preceding formulae. By this we merely reverse the order of the occurrence 

 on the major or on the minor radius, and thus the general condition of the con- 

 tact of two loops is contained in the proportion 



a : jS : : tan aT : tan /5T ; 



in other words, we have to find two arcs aT and /3T in the ratio of a to (3, and 

 having their tangents in the same ratio. 



7. If we put x for the tangent of the arc T, the above proportion becomes, 



a a a, — la — 2 o . 



a. a — 1 2 a. a. — la — 2 « — o 4 



1—1 9— X + T yj— o A X — &C. 



1 2*^1 2 



/3 /5 /3— 1 j8— 2 3 



f ^ — 1 ~2 3~~ ^ + &c - 



1 — 1 2 ^ "r 1 2 3 4"^ lV0 - 



On developing and subtracting the common term a(3z from each side, the result 

 is divisible by x z , and we obtain an equation into which only the even powers of 

 x enter. When a and (3 are prime to each other and both odd, the order of the 

 equation, x 2 being regarded as the unknown quantity, is ^ (a + (3 — 4) ; and 

 when one of them is even the equation is of the order ! (a + (3 — 3) ; and it is to 

 be observed that when a + (3 is even there is always the solution T = 90°, B = A, 

 which corresponds to a contact at the centre of the epicycloid. 



8. A table of the values of T and B (A being taken as unit) for all cases up to 

 (3 = 10, is subjoined. These values were readily obtained by the process which I 

 published in 1829 (Solution of Algebraic Equations of all orders) ; the values 

 of x 2 having been taken to ten places of decimals ; and those of T and B having 

 been thence computed by the ordinary seven-place tables. 



It may be noticed that the only case in which the ratio of A to B can be ex- 

 pressed by integer numbers is that of a : /3 : : 1 : 5 ; this rationality being con- 

 nected with the fact that 1 and 5 are the only two component parts of the perfect 

 number C which express a ratio in its lowest terms. And farther, that the ratio 

 of A to B is the same whether the arms turn in the same or in opposite directions ; 

 this fact is exhibited in the accompanying figures. 



