919 



INVERSION. 



INVOLUTE AND EVOLUTT. 



950 



the rest is as follows. Find the solutions of the equation a x = <f> x, 

 and let them be o> l x, a, x, &c. Then <J>-'.r being the convertible 

 inverse of <p x, the remaining inverses are u l $~ l x , w 2 <p 1 .r, &c. Thus 

 in the preceding example <p~ l x being the convertible inverse, the other 



is 2 - <t>~ 1 x. 



There is a remarkable class of 



[PERIODIC FUNCTIONS.] 



functions, each of which is its own inverse, such as 1 x, -, V (1 x*), 



x 



&c. Now if <t>x=<f>- 1 x we have tf>ipx-x, and these functions will be 

 considered, in the article just cited, as periodic functions of the second 

 order. 



The equation <t><f>~ 1 x = x being understood, suppose that between 

 the first and second operations we interpose the operation a, so that we 

 have 0a^>~' x. This is no longer equal to jc, but it is a function, the 

 properties of which are closely connected with those of ax. For 

 instance, if ax and &x be inverse to each other, then $ a <t>~ 1 x and 

 ~ l x are also inverse to each other: for af>x=x and <t>a<f>- 1 



' <p Q- 1 x, or<f> a j3 <t>~ 1 x, or $ -' x, or x. Thus 



knowing x + 1 and x 1 to be inverse functions, we know immediately 

 that log (* + 1) and log(t'-l) are inverse functions; and also V(- 2 + l) 

 and V (a: 1 !). For more detail on this subject see the article ' Calculus 

 of Functions,' in the ' Encyclopaedia Metropolitana.' 



INVERSION, in Music, is a change in the relative position of two 

 sounds, or of the several notes of a chord. Thus, c D, an interval of a 

 2nd, becomes by inversion (D c) a 7th. Example, 



I $>/3<f>-M 



And CEO, the notes of the triad, or perfect chord, by inversion 

 become the chord of the 6th (E o c), or of the -jth (o c E). Example, 



For other musical inversions, see CANON and FUQUE. 



INVESTITURE. [FEUDAL SYSTEM.] 



INVOLUTE AND EVOLUTE (the curve unrolled and the curve 

 from which it is unrolled), a name given to two curves so formed and 

 placed, that supposing the second to be cut out from solid matter, the 

 first can be formed by fastening one end of a thread upon a point in 

 the second, attaching a pencil to the other end, and moving the pencil 

 so that the thread may either gradually enwrap or be unwrapped from 

 the curve to which it is fastened. Thus the pencil in the diagram is 

 describing the involute of a circle, or the curve of which the circle is 

 the evolute. But the evolute of a circle is evidently a point. 



The following figure represents an ellipse with its evolute. If the 

 thread be fastened at b, wrapped over i a, and continued to A, it will, 

 CLB it unwraps from a ft, describe the arc A B ; and B A' while it wraps 

 over ft a'. If fastened in a similar manner at b', it will by the same 

 process describe A B'A'. 



If the line p V be drawn tangent to the evolute at p, it is one of the 

 position* of the thread, and P T, the tangent of the involute at P, is 

 perpendicular to p P. Also p r is the radius of curvature of the 

 involute at p; this is to say, no circle can pass so near the curve at r, 

 sui the one which has p for its centre and /> p for it* radius. [CURVA- 

 TURE.] Also, any arc of the evolute is the difference of two radii of 

 curvature of the involute : thus the arc tip is the difference between 

 a A and p v. Such are the principal geometrical connections of the 

 two curve*. 



Every curve has one evolute, and an infinite number of involutes. 

 For instance, fastening the thread at b, and continuing it to M instead 

 of A, we may with the cheeks a 6 and b a' produce another involute 

 from them (represented by a dotted line) ; and any number, however 

 great, by varying the position of M. But none of these involutes will 

 be ellipses, except the one from which the evolute was made ; though 

 they will all be ovals having remarkable analogies with the ellipse. 

 The proper name for curves described from the same evolute is 

 parallel curves, since they have the fundamental property of parallel 

 lines : for they never meet, though (if they admit of it) ever so far 

 produced ; a straight line perpendicular to one is always perpendicular 

 to the other ; and the part of the perpendicular intercepted is always 

 of the same length. When arcs of parallel curves are required to be 

 laid down, the most commodious method of proceeding is to construct 

 the evolute of one of the arcs approximately, as follows. On the arc 

 draw tangents at moderately small distances, and draw perpendiculars 

 to those tangents. The parts of the tangents cut off from each by its 

 neighbours will together give the arc of the evolute near enough for all 

 purposes. And it may be well to notice that it will be a sufficiently 

 accurate method of drawing the perpendicular to the tangent at a point 

 p, if we take a small circle whose centre is P, bisect the arc A c B in C, 

 and join and produce p a 



11 



The angular error thus committed is only a small portion of the 

 angle made by the tangents at P and A. 



Whenever the two arcs adjacent to a normal (or perpendicular to the 

 tangent) of the involute are equal and similar, there is a cusp in the 

 evolute ; and the evolute generally recedes without limit as we approach 

 a point of contrary flexure in the involute. 



The mathematical method of finding the evolute is as follows. Let 

 y=<t>xbe the equation of the involute, and let x and Y be the co- 

 ordinates of the point on the evolute corresponding to that on the 

 involute whose co-ordinates are x and y. Form the three equations 



lI)-i^-*>=< 



and from them eliminate x and /. The resulting equation between 

 x and Y is that of the evolute. But if the evolute be given, and the 

 involute is to be determined, let Y=y x be the equation of the former, 

 and from this and the latter two of the preceding three equations 

 eliminate x and Y. There will result a differential equation of the 

 second order between y and .r, among the primitives of which is the 

 equation of the involute. But the differential equation of the involutes 

 is one of the singular primitives of this equation of the second order, 

 and the question is most easily treated in the following way. Find the 

 differential equation of the curves which cut all the tangents of the 

 given evolute at right angles : those curves are the involutes required. 

 Thus if the curve be a parabola having the equation y=c 3?, the 

 equations for determining the evolute are 



x + 2cx(v y) = 0; 

 2c(l y) = 0; 



from which we find 



\ 



which give Y = -L+JL(|!.) S , 



