Is my book making an algebra error?

[ozone]

Not sure if this is a notation error or an algebra or if I am wrong , but my book presents a system of equations and I cannot get it into the form which it is presented

\tau _b = \left(1 + \frac{\Phi _b}{c^2}\right)\text{ }\text{$\Delta $t}

\tau _a = \left(1 + \frac{\Phi _a}{c^2}\right)\text{ }\text{$\Delta $t}

\tau _b = \left(1 + \frac{\Phi _{b-}\Phi _a}{c^2}\right)\text{ }\tau _a

Just wondering if I am missing something obvious here or if they are performing black magic. I tried subbing in for delta t and then doing long division but my answer was slightly different from their form.

 

[TSny]

Not sure if this is a notation error or an algebra or if I am wrong , but my book presents a system of equations and I cannot get it into the form which it is presented

\tau _b = \left(1 + \frac{\Phi _b}{c^2}\right)\text{ }\text{$\Delta $t}

\tau _a = \left(1 + \frac{\Phi _a}{c^2}\right)\text{ }\text{$\Delta $t}

\tau _b = \left(1 + \frac{\Phi _{b-}\Phi _a}{c^2}\right)\text{ }\tau _a

Just wondering if I am missing something obvious here or if they are performing black magic. I tried subbing in for delta t and then doing long division but my answer was slightly different from their form.

Could it be that they are making an approximation under the assumption that $\frac{\Phi_a}{c^2}$ is much smaller than 1?

 

[ozone]

Ahh I suppose that you are probably right.. thank you.