150 UNDULATORY THEORY OF LIGHT. 



Let us now sup])Osc that the second irapnlse, tliongli still normal to the first, 

 is not imparted either at the limit or in the middle of its path, but at a point 

 corresponding to 2-t, where t may have any value whatsoever. The velocity 

 of the body at the time t will bo aii\n2-t. The velocity produced by the 

 seoond impulse necessarily commences with the maximum: — , that is, with the 



velocity belonging to the time t='y--^ '^tl hence is «'.sin--, or tt'.siuDOO. 



Then the difference of the two, in respect to pJiasc — that is, to the degree of 

 their advancement in their respective vibrations — will be 2-;!— 90*^, or 90^ — 2-;;; 

 which, for convenience, put equal to 0. After a further time V, the tv*'o veloc- 

 ities will be — 



1. a. ^ai[2-t + 2-t') = a. sin(90O-l-(? + 2-^') = «- cos(2-^'_l-fl)=2/. 



2. a'. sin(90O + 2-i!') = a'.cos2-i;' = a;. 



Expanding y, and eliminating 2-t', there results the equation, 



a'hf+a'^x^—2aa'xy.(^o^0 = a'^a'hm^0. [1.] 



This is the general equation of the ellipse referred to its centi-e. It follows, 

 thereibre, that any two impulses, applied in directions at right angles to each 

 other to a body susceptible of vibration, will cause the body to describe an ellip- 

 tical orbit, whatever be the interval between the impulses. 



If, however, we suppose the second impulse to be in the direction of the orig- 

 inal vibration, and not at right angles to it; and, as before, that there is a differ- 

 ence of time corresponding to the arc 0, then the body will be impelled by two 

 conspiring or conflicting forces, capable of producing the simultaneous velocities, 

 a mi(f, anda'sin(9? + y). 



Let us put [a+a'ci^^Of+{a'iilnOf=^A^. 



^ ia + a'cmOf (a'amOf 2 , • 2 



Or, ^^ , ,, -— H- ^ -. „^- = 1 = cos^w + sm^w. 



A^ A^ 



the symbol w denoting a determinate angle. Then, 



a + a'cos^ = Acosw; and a' sin^ = Asinw. 



Let the first of these equations be multiplied by sin^?, and the second by 

 COS9P : their sum, added member for member, will be — 



a sin^-fa' sin^" costf + a' cos^ siu(9 = A sin^ cosw + A COS55 sinw. 



Or, a siu^ + a' sin(9? + tf)=A ^\n{f + u>). [2.] 



The first member of this equation is the expression for the velocity which the 

 body will possess at any time answering to (p, after the commencement of the 

 vibration a, which is least advanced in phase, and the second member shows 

 that this velocity is that which would exist in the body at the same time, had it 

 been acted on by one impulse only, capable of imparting the velocity A, and 

 applied at a moment earlier than a by the time corresponding to u>, and later 

 than a', by the time corresponding to O—m. 



The value of the velocity A, in terms of the original velocities a and a', may 

 be obtained by developing the assumed equation above, when it takes the form, 



A? = a^ + a'^- + 2aa'co%0; or, A = -l- yf d^+a''^+2aa'co'£,0. [3.[ 



This expression is remarkable, as being the value of the diagonal of a par- 

 allelogram, of which the sides are a and a', and the 

 angle of their inclination zzi 0. In the figure annexed, 

 let AB = DG = a, and AD = BO = a'. Also let BOD 

 = ADE^^. AE being drawn perpendicular to OD 

 produced, we have DE = a'cos6', and AE=a'sin/y. 

 Then, 



