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(a, b) = f a(s\)b{\t)d\, J o we have (a ± by c) = (a, c) ± (£, c), (a, b ± c) = (a, i) ± (a, c), ( 0 , &),<;) = (a, (6, c)) = (a, 5, c). From this follows readily the meaning of (a, 6, c, d). Putting, again, [a> b] = a + (a, b) + b, we have [0, a] = a, [a, 0] = <z, [[a, ô], c] = [<z, [b, c]] = [a, bt c]. We put finally, {a, fc, c} = 0 , £, c) + (a, b) + (&, c) + b. The function a is said to be reciprocable if there exists a function â such tha t (*) [a, â] = 0 and [â, a] = 0. (Each of these equations, it is well known, implies the other.) We say then that â is the reciprocal of a. If a is reciprocable, then so is <x, and the reciprocal of â is a. In what follows we shall designate the Fredholm determinant of a function a by Da, and the reciprocal of a by â. Of the various relationships that exist among the symbols (a, &), (a, b, c), [aj 6], [a, &, c] and {a, &, c\, we state here the following: (1) [a, J, c] = {a, J, c} + [a, c], (2) [a, b, â] = {a, &, â } . The following relations also hold true : (a) {a, 6, 0} = (a , &)+& {0, a, 6} = (a, 6 ) + a {a, 0, &} = 0 ,

On relations existing between two kernels of the form (𝑎,𝑏)+𝑏 and (𝑏,𝑎)+𝑏