422 Mr. atwood’s Theory for tie MenfuraUon 
Col. LU 
— 1 6 r pp X 1-2 — rri' + r 2 m + 2 nr p z — a. m z n z p z -f 2 n z rn ? f 
m' rip x */ £ _ ,r x V 1 - ■ r x 3 — 4 p z 
^ 1- p z 
5 0 • <7 <7 9 o »>' O' r J ') r ) ' 7 V 
p s s X I — 2 s p“ — m + n ' rti -j- p~ >n — 2 £ ttT h + 2 p~ trt tf V 
m 
i np x*y 1 — « ’ x v 1 — /> 2 x 2 — $ 
+ 1 b z fmnX 1 - n-f + 2p z n 2 -i z p z n z ~-2pr.X ^ \ -* 2 x ^i-rX' 7 ]-/ 
+ 1 X — n + ip n — s p n 
^ i—s z x v 1 
■p 2 xpx I — 2h Z 
v 1 - » 2 
27. This value of col'. ED is expreffed in terms of the va- 
riation of the fines of the given quantities : if it be neceffary 
to exprefs cof. ED in terms of the variation of the arcs them- 
felves, it muff firft be confidered to what part of the quadrant 
they belong : for example, if s be a line of an arc b not very 
near the extremity of the quadrant, and the variation be s, the 
cotemporary variation of the arc b will be p—=== z 5 but if the 
variable arc be nearly = 90°, and becomes exadfly equal to it 
ultimately having varied by a fmall arc b of which the verfed 
e * • . 
line = v ; then will - s = the verfed line of b and - b = v 2 V. 
Laflly, if the variable angle approximates to 90°, but is not 
equal to it, and the variation of its fine fihould be = r, the co- 
temporary variation of the arc muff be obtained from the 
general theorem in art 25. When either of the two lat- 
ter cafes happen, the variation of the arc muff be 
determined for each particular cafe ; but it will be neceffary 
to give a general expreffion for cof. ED in terms of the varia- 
tions of the given arcs, of which/, r, m , n , are the refpedfive 
fines when thefe arcs are at fame diffance from 90° - r this is 
contained in the next article. 
