128 



Proceedings of the Royal Irish Academy. 



On substitutiug iu the contour integral, the first term integrates to zero ; the 

 asymptotic value of the integrand arising from the second is \'hl\f^{t); 

 the integral is therefore 2TTif^(t). Thus the result has been established. 



8 4*3. TJie solution (57) satisfies the differential equations if the elements of the 

 altered roio in the first determinant are rei^laced, hy any 'polynomials 

 vjhatevcr. 

 The proof is as in § (4'2) and the beginning of § (-S-l). 



^ 4*4. The solution (hi) gives as initial values of x, y, ajtid their derivatives v/jj to 

 D'"-^(x), D"'\y), the constants vjhicli occur in the first column of the first 

 deter miTUint, provided these are compatible with the differential eqiiations. 



When A is "abnormal" the present discussion pays no regard to, and, indeed, 

 fails to settle, the number of independent constants in the general solution. 

 From other discussions it is known that this number is the number which 

 indicates the order of A, and thus, that, when A is " abnormal," not all the 

 m + n initial values of x, y, and other derivatives can be assigned arbitrarily. 

 It is in keeping with this fact, known otherwise, but not assumed here, that 

 I cannot prove that the solution (57) will necessarily give as initial value of 

 DPx (say), p < m, the letter, oTp, which indicates it in (57), but merely that it 

 gives a combination of the letters denoting initial values which is equal to 

 iTp if those letters in the formula (57) actually represent values compatible 

 with the equations which are to be solved. 



On writing the determinant in the second term in the right-hand member 

 of (57), by the aid of (20), in the form^ 



«>ii(A)r eH^-nf,{tyit' + <P.,^(\) ' e^{t-nf^(f)dt' 

 Jo "Jo 



■AlO. 



(60j 



differentiating (57) under the integral sign, and again using (20) to transform 

 further the part obtained from the second term, we obtain 



^^!0iiW -»n(A)i.i'+ {»i2(^) - ■/■i2(A)Jy 



2iriD!'x = 



U>^idX 

 c^'dX 



A(A) 



D-X 



>pJX} 



1)-X 



f22 



X] 



t = o 



ft 



XP^^^{X)\' eHi-nf^{t')df + X^<I)2i(A)l cKt~t')f[t\dt' 

 Jo Jo" 



^ D^^rmji]^M^f^it) + ^'*2i(-^^ ->"i'2i(A)_^^^^y 



IJ -X 



D-X 



(61) 



' Here, again, I introduce the minors, to use notation suitable when there are more 

 than two unknowns. 



