410 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
If we add these equations together, 
+ O'// + O/;/ + (o — O////) = ( q, 111^111 ~ I, — ^ (408.) 
If we multiply the first of (407.) by 2®, the second by 2^ the third by 2, and the 
fourth by 2°, and add the results, 
2V~<r;///=(^////r,^+^///r,/+2^,,r,+4^,r-85')J-^+('^,„+‘^^,^+2''F,+4^-8tI), . (409.) 
an integral which enables us to approximate with ease to the value of the integral of 
the third order and circular form, in terms of an integral of the first order. 
We have shown in XXVIII. how the integral of the first order may be reduced. 
The above expressions may be reduced to simpler forms, when the functions are 
complete. In this case 0=0, T=0, T^=0, T,^=0, &c.; and when j is a quadrant, 
will be two quadrants, <7^^ will be four quadrants, will be eight quadrants, and so on ; 
the preceding expression may now be written, denoting a quadrant by the symbol ?, 
lb(^-^/J = (7///i^/«+9;//^;;+2?//^+4g/r-8y)J^ (410.) 
In (396.) we found for the parameter of the derived integral of the third order, the 
expression Or, referring to the geometrical representatives of these e.\- 
pressions, we found for the focal distance f, of this derived curve, the expression 
«,= tan%=^j-q;;^ ; but if we turn to (389.) we shall see that this is the expression 
for the maximum protangent to the original spherical ellipse, which is given by the 
equation tanV= 
mn 
We thus arrive at this curious relation between the curves 
successively derived, that the maximum protangent of any one of the spherical ellipses 
becomes the focal distance of the one immediately succeeding in the series. 
LXXIX. Given m, n and i, we may determine m^, and 
1 7 77777 
for Substituting these values of and n, in the equation which 
connects the parameters, — ni-\-mp,,=P, 
"*/=[■ 
V' 1 +77— a/ 1 —777 
1 +77 + 1 ■ 
~I 2 
— m I 
— m-J 
( 411 .) 
Hence given m, n and i, we can easily compute the values of m^, and i^, and then of 
m,,, rill and in ; and so on as far as we please. 
Given the semiaxes a and b of the elliptic cylinder, wdiose intersection with the 
sphere is the original spherical ellipse, to determine the semiaxes a, and of the 
cylinder, whose intersection with the sphere shall be the first derived spherical ellipse. 
We may derive from (53.) and (54.) the values of a and b in terms of m, n and /, 
or eliminating i, in terms of m and n only. Now 
u ”/ ^_n ,(l-m ,) 
m{\+n) m ‘ ^ k^ 7ft^(l+n,)’ m. 
