View of the Undulatory Theory of Light. Ill 



Again, if the quantities ABC be so chosen that on multi- 

 plying the first term of each member of the equations (22.) 

 by A, the second by B, and the third by C, we have 



LA + RBh-QC_ RA + MB + PC _ QA + PB + NC,_. 



-^ ^b~"~~ - C ^^^'^ 



and write these equal to s*, then, on differentiating the equa- 

 tion (26.)jand substituting the values from equation (22.), we 

 shall find the second differential coefficient to be the remark- 

 able form dU _ 3 . 

 'TP ~ ~' ' ^^^'^ 

 From (27.) it is evident that we have three values of s^ cor- 



R C • 

 responding to three systems of values of the ratios -^, —t- ' 



and consequently there are three straight lines with either of 

 which the line O A may coincide ; and the same equations en- 

 able us to determine these lines, for they evidently coincide 

 in form with those mentioned in the preliminary article. We 

 can deduce the same equation of the third degree, which for 

 reference we will call (29.); and consequently the three lines 

 OA'OA"OA'" are identified with the three axes of the 

 surface of the second degree represented by the equation there 

 assumed, involving as coefficients the quantities L M N, &c., 

 and which we will call (30.). If it be an ellipsoid, the three 

 values of 5^ are equal to the squares of the three semiaxes of 

 the ellipsoid. 



These considerations then enable us to assign the displace- 

 ment of m at the end of the time /, in directions parallel to 

 three determinate lines at right angles. Let these three dis- 

 placements be expressed by accenting the letters in equation 

 (26. )> or let us suppose 



«' = A'f + BS + C? ^ 



8" = A"f + B"), + C"? L (31.) 



«'" = A"'f + B"'>3 + C"? J 



In each set of the coefficients the relation (24.) holds good ; 

 also, since the lines are at right angles, we have 

 A'A" + B'B" + C'C" = ^ 

 A"A'"+ B"B'"+ C"C"'= 

 A' A'" + B'B'" + C'C" = 



Hence we deduce from (31.) 



^ = X'b + A" 8"+ A'" 8'" 



rj = B'8' + B"8" + B"'8"' y (33.) 



(32.) 



