492 



METHOD OF HUMEISCAL INTEGRATION OF THE EOUATIOHS OF MO TIP H 



InSLiection of equation (6) shows that while the dspenOsnt and inoecendent variaDles can 

 easily De seoarated, making solution by numerical quadrature ocssi&le, the right hand side has 

 a zero at the start of the integrdtion (the origin of time is chosen at the Instant of minimum 

 radius). The integrand In the quadrature orocc-ss is therefore infinite at the start, and the 

 singularity is not easily removed. Accordingly the following process of numerical integration 

 was adODteO. 



The first five or six values of a at equal intervals of t were comcuted By writing down 

 the first few terms of a Taylor series fora near the minimum value a^. Tne first and third 

 derivatives of a with rssoect to t vanish at tne origin. The second and fourth derivatives wer 

 calculated by reoested differentiation of the right hand side of (6), making use of the 

 condition (7) to remove the carametsr m. The following values were jDtalned: 



= -i- a,-' (1- i ca,-*) 



u n 



(1 - i c.-') [l - ^ caf ^ 



The time interval for each step' was cnosen so that dfter about six steos the term in t 

 was still stiBll. The Taylor series was also dl f fenent iated and the resultant series evaluated 

 at the last time interval. This value was then comoared with the value of da/dt calculated 

 directly from equation (6) with the acproorlate value of a Inserted to verify that neglect of 

 higher terms in the Taylor series had causeo no aaoreciable error. 



With these starting values for a, and the corresponding values of da/dt calculated from 

 (6) the process described on page 9H2 of ° Interoolat ion and Allied Tables* for Integration of 

 differential equations vf first order was carried out'. The fourth differences of da/dt were 

 watched but at no st?.ge was their c.^ntribut l:n t: the Integration aopreciable. 



"Interpolation an:; AUie; Tables" oublisheo by H.M. Stationery Office .n behalf 

 of H.M. Nautical Alm,->.nac Office. 



