This is the example 6.2.19 of Statistical Inference Book by Casella and Berger.
Scale parameter families also have certain kinds of ancillary statistics. Let $X_1,...,X_n$ be iid observations from a scale parameter family with cdf $F(x/\delta), \delta >0$. Then any statistic that depends on the sample only through the $n-1$ values $X_1/X_n,..., X_{n-1}/X_n$ is an ancillary statistic. For example,
$$\frac{X_1+...+X_n}{X_n}=\frac{X_1}{X_n}+...+\frac{X_{n-1}}{X_n}+1$$ is an ancillary statistic.
To see this fact, let $Z_1,...,Z_n$ be iid observations from $F(x)$ (corresponding to $\delta =1$) with $X_i=\delta Z_i$. The joint cdf of $X_1/X_n,..., X_{n-1}/X_n$
$$F(y_1,...,y_{n-1}|\delta)=P_\delta (X_1/X_n \leq y_1,..., X_{n-1}/X_n \leq y_{n-1})=P\delta (\delta Z_1 / (\delta Z_n) \leq y_1,..., \delta Z_{n-1} / (\delta Z_n) \leq y_{n-1})=P_\delta (Z_1/Z_n \leq y_1,....,Z_{n-1} / Z_n \leq y_{n-1})$$
The last probability does not depend on $\delta$ because the distribution of $Z_1,...,Z_n$ does not depend on $\delta$.
I can understand the above, which is clear. But I am confused that the statement "Then any statistic that depends on the sample only through the $n-1$ values $X_1/X_n,..., X_{n-1}/X_n$ is an ancillary statistic." Does this mean these $n-1$ values need to show up at the same time?
For example, if $n=4$, is $X_1/X_4, X_2/X_4$ an ancillary statistic?