OF PRESSURE AND EQUILIBRIUM. 93 



may write A and h for A' and Ji\ in tlie same sense. The 

 initial value of u has sometimes been distinguished by a 

 capital letter (Phil. Trans. 1819); Mr. Wronsky marks it 

 by a point, ii, at least when u is supposed to vanish ; but 

 we must not altogether forget that this is the Newtonian cha- 

 racter for a fluxion ; the point, if it were thought neces- 

 sary, might be written under the letter, m, or a prosodial 

 mark might be employed instead of it, as m, or rather m, 

 which would partly explain itself; as indeed u may be said 

 to do. 



Corollary 1. If A be an arc =5 and u its 



.• ' A , dtt' ^ d^w' 



sine, making M z=0, we have -77= cos 6'r=l; 7-7"— 



A d^M . , . 1 



— sm «=U; -7--= —1..., whence sin 5=5— TTr«* + 

 as-^ 2.3 



2..5 



Corollary 2. In the same manner cos 5zzl — 



Corollary 3. If u—a"", since -r-zia^hla (51), 



d^M 



j-^i=a%l2a, ...; putting jc'izO, and a"^ zzl, we have 



a^=zl+hla.a; + hl2a. ---4.hl3a.r-:r+ ... 

 1«J 1..0 



^ . ,p , . ^ dw' 1 d^w' 



Corollary 4. If u—h\(a-\-x\ t~=-, - — = — 



1 d^u 2 ,^,^ , ,, X ^2 j;3 



^'d^=^----' ^"^^^^^^ + ^)=^^^ + ~2^ + 3^— 



Hence hi (l-\-x)^X'- ^-f- •-»— . . . ; and hl(l — x)— — x— 



x^ x^ ,A+x ^ ^ x^ x^ 



-5-— -g— ...; consequently hlr-— :=:2 (a:'4"^ + T". • Oi 



