81 
then f > 
dr 
J;{U„ + U_l + U_2 + &0.} +p{Vo + V_l + V_2 + &C, 
‘1] 
sinp(at -r) + r|;(Vo + V_i + V_2 + &c.} -i>(U„ + U_i + U_a 
+ &c. } jcosp(at - r) = r[ { + -3 + | sin/)(at - r) 
+ {'y_i + v_2 + v-3 + &o-}cosp(at-r)] 
Whence 
tu_i=pYq ; -pUo (2) 
dYo tt 
rv _2 = - 5 - ; 
ru_i 
dr 
dr 
&c. 
r^^_2= — r- +i5V_i 
dr 
dY_r 
dr 
&c. 
-; 9 U. 
Note. — To this change from u_i to Yo and from v^i to 
Uo corresponding to an addition of ^ to the phase, is given 
the name of the loss of a quarter undulation. 
Whence if u, v, &c., are known and the limits of r are 
known, U and Y can be found ; for instance 
p[^A = [ru_,t\ 
'Id 
p dr 
{ru_i 
) 
Having thus found [^1] , we may complete the solution 
and find 
where and are the limits of r, a and ft those of 0 . 
II. 
In this way we may consider the direct problem to be 
formally solved, but for the purpose of this paper we need 
to know in what way the several terms capable of appearing 
in will affect the several terms Yq, U_i, &c, 
