1890 - 91 .] Lord M‘Laren on Glissette of Two-term Oval. 
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Equation of the Glissette of the Two-term Oval — + T- - 1 , 
a n b n 
and Cognate Curves. By the Hon. Lord M‘Laren. 
(Read January 19, 1891.) 
The first step is to find an expression for the distances A., /x, of 
the centre of the oval from the guides. These distances are to he 
found separately, beginning with A. 
For this purpose it is indifferent whether we consider the oval as 
moving in contact with the guides, or whether we consider the 
guides as variable tangents moving round the circumference of the 
oval. On the latter supposition, the extremity of A is evidently the 
locus of the foot of the perpendicular on the tangent of the oval, 
whose equation is given in the title. 
Let £, rj he the coordinates of this locus referred to the principal 
axes of the generating curve as reference lines. Then, by a known 
relation, the equation of the required locus is 
(£ 2 + y 2 ) = («£) + (by)^! . . . (A). 
The distance, A, is the radius-vector corresponding to £ and y : 
therefore £ — A cos 6 ; y — Xsinb. 
Substituting these values in (A) we have 
2 n n 2ra n n n 
A M_1 (cos 2 0 + sin 2 6) n ~ x = A”" 1 = A n_1 {(a cos 0) n ~ l + ( b sin 0) n ~ l } , 
whence 
n n 
A M_1 - (acos 0)"- 1 + (6 sin 0) 71-1 ; . ... (1) 
Similarly, 
n n n 
/x n ~ 1 = (a sin 0) n ~ x + (b cos 6) n ~ x ; . . . . (2) 
(by interchanging sin 6 and cos 6). 
Next, let the coordinates of the tracing-point ( i.e ., its distances 
from the guides) be denoted by X, Y. 
X is found from A in the same way as in the case of the glissette 
of the ellipse, as given by Professor Tait ( Proc . Roy. Soc. Edin., 
vol. xvii. p. 2). 
a and r (constants) are the polar coordinates of the tracing-point, 
