When presenting a series of displayed equations without intervening text, do not add commas to the end of each one if they form a sequence:
We find that
\begin{aligned} f(x)&=(x-1)^2+(x+2)^2 \\ &=(x-1)(x-1)+(x+2)(x+2) \\ &=x^2-2x+1+x^2+4x+4 \\ &= 2x^2+2x+5 \end{aligned}
Note that these are aligned on the ‘=’ sign.
When presenting a list of equations not derived from each other, use commas:
Earlier results include
\begin{aligned} \psi_1(\theta)&=\frac{(m-1)}{p}\psi(\theta),\\ \zeta(s)&=\sum_{n=1}^\infty\frac{1}{n^s}, \\ f(u)&=\sqrt{1-c^2\phi^2(u)}. \end{aligned}
There is no need for and before the last equation, but it can be included.
When presenting many definitions of symbols, consider presenting them as a series of equations, rather than a bulleted list:
As is conventional, we define:
\(y_{i,j}=\) the observed result of using treatment \(i\) on cohort \(j\),
\(T_{i\cdot}=\) the sum of the observations for the \(i\)th treatment,
\(T_{\cdot{}j}=\) the sum of the observations for the \(j\)th cohort.
