Miscellanies. 393 
And by the well known series, 
tan. — tan. °~-3-+ tan. °+5 —tan.7~7+ &c. = the corresponding 
arc, we have 2rnp—(n? — p*) —(2rnp)*+-3(n? — p?)* + (2rnp)*— 
5(n?—p?)'—&e.=9. i 
Observations.—1st. If n=p, then g=one quadrant. If*m or 
p=0, then op=0. 2d. If n>>p, then the functions of 9, indicated 
‘by the expressions are all positive quantities. If p>, then the co- 
sine, secant, tangent and cotangent will be negative quantities. In 
this case r(n? — p?) =q is an absurd and impossible expression, but 
we may put 7(n?—p?)+(n?+p?)=-—gq. So also of the other 
functions enumerated. 3d. When a sine, tangent or secant of 9, or 
its complement=a, the values of two of the quantities which enter 
into the expressions, viz. of 7, or p, may be assigned arbitrarily, 
within certain limits, =6, ¢ or d, and, the other being =a, may be 
found by an equation not exceeding a quadratic, and if the conditions 
the last observation be preserved, there will be no imaginary expres- 
sion. 4th. The expressions for the sine, tangent and secant of 9, 
and its complement are each comprised in a finite and small number 
of terms, which have no radical or surd quantities. If the value of 
any one of these functions be assumed equal to a given-quantity, then 
indeed, the value of one of the quantities may be inexpressible in 
numbers except by an infinite series, or a radical quantity, but the 
’ . general analytical expression will always be entirely free from radi- 
cals. 5th. The expressions cannot possibly transcend the limits 
of the values of the functions they represent. Thus it is impossi- 
ble to make 2rnp+(n?+p?)>r, or r(n?—p?)—+(n?+p*)>r, or 
r(n?+-p?)+2np<r, or r(n?+p?)+(n?—p*)<r. 6th. Putting 
r? =a?, the expressions give a formula for finding two numbers, the 
sum or difference of whose squares is equal to the square of a 
given number. For sec.? —tan.?=rad.*, and sin.*+-cos.? s=rad.?< 
7th. The arbitrary quantities, mentioned in observation 3d, may be 
used for the purpose of making the expressions conformable to any 
two possible conditions in addition to those there mentioned. 
eries.—1st. May not the expressions given in this article be 
used in some analytical investigations respecting the circle and its 
functions" in preference to those generally adopted? 2d. Is it possi- 
ble by any modifications of these expressions to represent the rela- 
tion of to its common functions in a limited number of terms? 
