DR WALLACE ON A FUNCTIONAL EQUATION. Q^i 



and hence, by integrating, and substituting for a . /3 &c. their numeral values 



(14) 



^3 5(f,^ 61 (p' 1385 <p^ 



'^~^"*" 1.2.3 ^ 1.2. 3. 4. 5 "'"1.2. 3. 4. 5. 6. 7 "'"1.2. 3. 4. 5. 6. 7. 8. 9"^'^°" 



x^ 5x^ 61 z' 1385 3~> 



f'-''~ 1.2.3 "''1.2.3.4.5 1.2.3.4.5.6.7 "^1 .2.3 .4.5. 6.7.8. Q"^"' 



44. In the application of these formulae, it must be remembered that <p is ex- 

 pressed in parts of the tabular radius of the trigonometrical tables : therefore, if 

 the angle be expressed in minutes, it must be multiplied by the number 3437.74677 

 (the radius reduced to minutes). If the angle (p be considerable, the series will 

 converge too slow to be useful. 



A convenient expression, as an approximation to the value of x, may be 

 found from the series by the following process : We found that 

 , d)' 5(p' 610' 



■ ^ ^ (P^ (t>^ 4>'' n 



Now «'-'^=<^-T-"'m-5Sio + ^'=- 



. ^ ^ 2(±' 16d)^ 272(4' , 



rj,^ !> . J. ■ ^ 3(4= 15(p' 273d)' „ 



Therefore tan fP - sm <^ = ~^ + -^^ + --~^- + &c. 



, /. ^ • ^^ 0' 5d)' 91 (i' „ 

 and i(tan.^-sm<^)=.-|-+-|^+-^-^+ &c. 



By subtracting the sides of this last equation from those of the ftrst, and trans- 

 posing, we have 



':=(p + i (tan (p-sm ^)-j^ &c. 



If the angle (p be not very great, we have, as an approximation, putting a for the 

 parameter, 



x= a { (p + ^ (tan cp — sin (p)}, (15) 



This in many cases may be sufficiently near to the value of a;. 



Suppose, as an example, that a=100 feet and 0=42°; the calculation will be 

 as foUows : 



cp (in parts of radius) =.7330388 

 From tan =.9004040 



Subtract sin (p = .6691306 



Divide by 3)2312734(.0770911 



.81013 ; . 

 «=100 {0 + ^ (tan 0-sin0)}=81.O13 feet. 

 The more correct value of x is 80.916 feet ; the corresponding value of y=a sec 

 is 134.563 feet ; and the catenary arc =a tan (p ~ 90.040 feet. 



