API Design for a Microtouch Touch Screen Input Controller

This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1893 edition. Excerpt: ...in question in that equation and add the equations and then investigate the limiting form approached by the resulting equation as p is indefinitely increased. As 12 is indefinitely increased the summation in question will approach an integration; and since d, a dcents =--sin 0.d0 dcents is the element of surface of the unit sphere, and as the limits--1 and 1 of /4 correspond to vr and 0 of 0 the integration is a surface integration over the surface of the unit sphere. In determining any coefficient as AM in (1) the first member of the limiting form of our resulting equation will be In the second member we shall come across terms of the forms where Y, (;i,4) and Y, (, u., cents) are Surface Spherical Harmonics of different degrees. If we are determining a coefficient B, _, the only difference is that sin ncents and cos 12cents will be interchanged in the forms just specified. 105. The integral over the surface of the unit sphere of the product of two Surface Spherical Harmonies of dzferent degrees is zero. 2. Show that the integral over the surface of the unit sphere of the product of two Tesseral Harmonics of the same degree but of different orders is zero. by integration by parts. _ Replacing n by n--1 in equation (2) Art. 84 and remembering that and the development holds good for all values of /L and cents corresponding to points on the unit sphere, provided only that the given function satisfies the conditions that would have to be satisfied if it were to be developed into a Four1'er s Series. If we use, u., and cents, in place of /4 and 4 in (1), (2), and (3), we can write (4) Formulas (1), (2), (3), and (4) are convenient for actual work; (5) is rather more compactly written. 108. As